dynamics and analysis of alignment models of …technical ingredient will be an adaptation of the...

DYNAMICS AND ANALYSIS OF ALIGNMENT MODELS OF COLLECTIVE

BEHAVIOR

ROMAN SHVYDKOY

Contents

1. Preface 22. Emergent phenomena of collective behavior. 43. Agent-based alignment systems 63.1. Types of communication and collective outcomes 63.2. Momentum, Energy, and Maximum Principle 73.3. Connectivity and spectral method 83.4. Alignment with heavy tail communication 103.5. Stability 123.6. Singular kernels and the issue of collisions 143.7. Degenerate communication. Corrector Method 153.8. Multi-flocks. Clusters. Multi-species 193.9. Notes and References 224. Forced systems 244.1. 3Zones model. Small crowd flocking 244.2. External confinement. Hypocoercivity 264.3. 2Zone model: attraction + alignment 294.4. Dynamics under self-propulsion and Rayleigh friction 344.5. Notes and References 365. Kinetic models 375.1. BBGKY hierarchy: formal derivation 375.2. Weak formulation and basic principles of kinetic dynamics 385.3. Kinetic maximum principle and flocking 405.4. Stability. Kantorovich-Rubinstein metric. Contractivity 415.5. Mean-field limit 435.6. Notes and References 456. Macroscopic description. Hydrodynamic limit. 466.1. Multi-scale model and its justification 476.2. Hydrodynamic limit. Kinetic relative entropy 506.3. Notes and References 557. Euler Alignment System 577.1. Basic properties. Energy law 577.2. Hydrodynamic flocking and stability 587.3. Spectral method. Hydrodynamic connectivity 607.4. Topological models. Adaptive diffusion 627.5. Notes and References 678. Local well-posedness and continuation criteria 688.1. Smooth models 688.2. Singular models 72

Date: February 13, 2020.

1991 Mathematics Subject Classification. 92D25, 35Q35, 76N10.Key words and phrases. flocking, alignment, fractional dissipation, Cucker-Smale.

Acknowledgment. Research is supported in part by NSF grant DMS-1813351 and Simons Foundation.

1

2 ROMAN SHVYDKOY

8.3. Notes and References 789. One-dimensional theory 809.1. Smooth kernels: critical thresholds and stability. 809.2. Entropy. Csiszar-Kullback inequality. Distribution of the limiting flock 819.3. Alignment on T with degenerate kernel. The corrector method 859.4. Singular models: global wellposedness 919.5. Notes and References 9710. Global solutions to multi-dimensional systems 9910.1. Unidirectional flocks and their stability 9910.2. Mikado clusters in hydrodynamic multi-flocks 10410.3. Spectral dynamics approach 10510.4. Nearly aligned flocks of singular models: small initial data 10810.5. Notes and References 113References 114

1. Preface

Collective phenomena occur in nature, technology and social behavior in a vast variety of contexts. Forexample, birds and fish use their sensory abilities to communicate and align their positions to form cohesivecongregations – flocks and schools, respectively. Migration of microorganisms such as bacteria or cells followprimitive bio-chemical communication rules to induce collective motion which is crucial to sustain live inmore complex organisms. In technology, communication between agents to achieve common collective goalsis the problem that spans across many different applications such as control of unmanned aerial vehicles,coordinated satellite navigation, traffic control, etc. Social science offers another variety of examples ofcollective behavior such as dynamics of opinions and reaching consensus, social and economic networks,emergence of leaders. With the advent of new techniques and methods in PDEs analysis of mathematicalmodels of collective motion is starting to become one of the most actively developing subjects of appliedscience. This book is a modest attempt to introduce the reader to one special class of alignment models,called Cucker-Smale system and its kinetic and hydrodynamic counterparts, which has received tremendousdevelopment in mathematical literature in recent years. From purely analytical prospective, one of the mostfascinating aspects of these system is that they bring together a toolbox of several modern techniques thatemerged recently in fluid dynamics, fractional analysis, and kinetic theory. Our primary goal will be to walkthrough all levels of description – discrete, kinetic, and hydrodynamic – in a mostly self-contained manner,and to highlight essential analytical aspects on each step. Those include regularity theory and long timebehavior of solutions. For the most part, we will be concerned with a “bare bone” model where the mainstructure is stripped of other accessories such as external forces, various modifications, etc, except for aspecial class of classical potential and friction forces due to their close relevance to applications.

The material of this book represents a modified and largely streamlined content of several works, manyof which are very recent and are still in production. All relevant references and discussions will be providedat the end of each chapter. We will by no means attempt to provide a comprehensive coverage of thetopics selected, but the reader will find these surveys extremely useful for further guidance on the subject[1, 13, 15, 97, 72, 69].

Our starting point is the agent-based Cucker-Smale system which we introduce in Chapter 2. We willbriefly discuss historical context and a range of relevant applications. A rough classification of collectivephenomena and most frequently used types of communication will be given. The basic question here is howmuch communication is necessary to induce unconditional alignment? We present two methods of approachto long time behavior – the Cucker-Smale original spectral method and Lyapunov function approach of Haand Liu. These methods will reemerge in both kinetic and hydrodynamic contexts later in the text. Afirst major deviation from the classical non-degenerate kernels will be provided with the introduction of thecorrector method, which allows to include more realistic communication protocols, for example, as a part ofmulti-zone models to be discussed in Chapter 4. We give a brief overview of existing multi-scale models atthe end of Chapter 3.

DYNAMICS AND ANALYSIS OF ALIGNMENT MODELS OF COLLECTIVE BEHAVIOR 3

Although in most of the text we focus on the core alignment mechanisms omitting any forces, in Chapter 4we provide discussion of a few forced models due to their relevance to applications and recent revelations inunderstanding their role in alignment dynamics. The central case study here is potential repulsion/attractionforcing and multi-zone analysis that comes with it. As in the degenerate kernel case, the energy law inpresence of the interaction force suffers from lack of coercivity. We present a new adaptation of the hypoco-ercivity method to tackle this issue. Dynamics under Rayleigh friction and self-propulsion in the context ofCucker-Smale alignment will be discussed in Section 4.4.

Analysis of kinetic alignment models – a Vlasov-type direct counterpart of Cucker-Smale system – will bethe subject of Chapter 5. The kinetic formulation will be derived formally via the BBGKY hierarchy andjustified rigorously through the mean-field limit. The analysis of the limit comes with additional contractivityestimates which in the case of heavy tail communication automatically show stability of flocking dynamics.The passage from kinetic to hydrodynamic description will be discussed in Chapter 6. Our approach is basedupon a kinetic version of the relative entropy method implemented to flocking models by Kang, Figalli andVasseur with a few modifications. In particular, we rely on a soft kinetic model which can be justified viamean-field, and cast the results for compactly supported flocks in the open space.

The hydrodynamic version of Cucker-Smale system, called Euler Alignment system, we will discussed indetail in Chapter 7. We revoke its Lagrangian description in order to draw similarities with the agent-baseddynamics. As a result, many number of agent independent statements carry over from the microscopicdescription to macroscopic description ad verbatim. We will introduce a class of topological models inresponse to deficiencies of classical metric models in the case when communication in local. Here thehydrodynamic connectivity goes into interplay with adaptive diffusion built into topological protocol toproduce natural flocking results. Subsequent analysis of such models will not be included due to technicalcomplexities that go beyond the scope of this text.

In Chapters 8 and 9 we provide the core well-posedness theory of both smooth and singular models.We start with local existence results supplemented by proper continuation criteria which will be useful todevelop global regularity theory for 1D and some multi-D systems. Understandably, the analysis of smoothmodels will be quite different from analysis of singular models due to the differences in the type of PDEs weare dealing with. Smooth models exhibit structure of a hyperbolic system of conservation laws, for whichwe can provide threshold criteria for regularity in terms of an entropy-like quantity. Singular models fallinto the class of fractional parabolic equations with rough drift, which enjoyed major development in recentyears in the context of fluid dynamics (critical SQG, fractional Burgers, etc). We will present a streamlinedproof of the main regularity result for 1D models introduced by Tadmor and the author in the thrilogy ofworks [90, 91, 92]. Here we recall relevant tools of analysis of critical fractional PDEs, making expositionmostly self-contained with the exception of Nash-Moser type Holder regularization result of Silvestre [93]which would carry us beyond the scope of the book. For the flocking analysis of singular models the keytechnical ingredient will be an adaptation of the non-local maximum principle of Constantin and Vicol [29].An independent approach to regularity via the use of modulus of continuity method appeared also in Doet al [39]. Although it is a powerful tool for studying fractional models we will not cover it for the sake ofbrevity, and refer to [39, 2, 60] for its applications to alignment models.

A few partial results are known on regularity of multidimensional Euler Alignment systems, however thetheoretical base in this case remains largely underdeveloped. We present them in Chapter 10. Those includesmall initial data for smooth models in terms of spectral gap of the initial condition, and for singular modelsin terms of amplitude of velocity oscillations. We reveal a new class of unidirectional solutions and provetheir regularity and stability in multidimensional settings.

Several pressing open problems will be highlighted throughout the text, and in the Notes and References.The material of this book is based upon several series of lectures given at ICMAT in November 2018,

Charles University in May 2019, and Beijing Normal University in July 2019. The author is grateful to JosefMalek for encouraging him to write and publish these notes in the Necas Center Series. Constant supportof NSF, Simons Foundation, and College of LAS at UIC during preparation of the manuscript is gratefullyacknowledged. Special thanks goes to Eitan Tadmor for introducing the author to the subject with opennessand enthusiasm he will never be able to match.

Chicago, USA Roman ShvydkoyJanuary, 2020

4 ROMAN SHVYDKOY

2. Emergent phenomena of collective behavior.

Generally speaking, emergence is a phenomenon of reaching collective outcome in a given system of agentswhich follow a prescribed protocol of communication. Some of the most striking examples of emergenceare pattern formations such as wedges of bird flocks, or milling motion in schools of fish, lattices in cellorganization or bee hives. These effects are achieved by following local rules of engagement that result inglobal outcomes over time. Depending on the context many mathematical models have been studied toreplicate a specific collective behavior. Roughly, most classes of models fall into two categories – 1st ordersystems and 2nd order systems. First order systems often model evolution of non-inertial flocks, those thatdo not induce motion if no force is present. For example, in studying exchange of opinions or networkingone of the most popular model used is environmental averaging:

pi = λ∑j∈Ni

aij(t)(pj − pi) + Fi,∑j

aij(t) = 1,

where Ni is a set of ’active’ agents in local proximity, and pi ∈ Rn stands for a characteristic state of i’thagent such as its opinion. The main alignment term represents statistic averaging over actively involvedagents and Fi is an external random or deterministic force to account for possible additional effects. Thetypical emergent phenomenon here is achieving consensus, pi → p, see [72, 37, 75] for further reading.

A large class of 1st order gradient models appears in numerous biological and physical applications suchas particle dynamics, cell migration, etc, is given by

(1) xi +∑j:j 6=i

∇xiW (xi − xj) = 0, i = 1, . . . , N.

where W is a radially symmetric repulsion/attraction potential and xi’s are spacial positions of the agentstypically on a given closed surface Σ ⊂ Rn or simply in Rn. The most well studied collective outcomes arelattice patterns in distribution of global minimizers of potential energy.

One famous example of a gradient system is given by Kuramoto synchronization model

θi =λ

N

∑j∈Ni

sin(θj − θi) + ωi, θi ∈ T1,

where θi are phase angles of agents and ωi are prescribed natural frequencies. It appears in surprisinglydiverse array of examples, such as neuronal signals in brain, simulating cardiac pacemaker cells, synchro-nization of power grids, and even cricket pitches in the garden, see [1, 61]. The model exhibits phenomenallycomplex behavior despite its relative simplicity. The size of coupling strength relative to natural frequenciesmay trigger phase transitions from chaotic to synchronous.

A class of models that take into account inertial effects are models of 2nd order describing evolution ofpairs of phase points

xi ∈ Ω ⊂ Rn, i = 1, . . . , N

vi = xi.

One of the most studied models in biological literature is the time-discrete Vicsek modelvi(k + 1) = v0

∑j:|xj−xi|<r0 vj∣∣∣∑j:|xj−xi|<r0 vj

∣∣∣ + Fi

xi(k + 1) = xi(k) + vi(k + 1).

Depending on the nature of the forces Fi solutions may form interesting flocking patters such as mills orperiodically rotating chains with or without self-intersections. The model undergoes phase transitions fromdisordered to ordered state, depending on the level of noise present in the system, see [97] for comprehensivediscussion.


In this book the center of our attention will be analysis of alignment systems, prototypical example ofwhich is the Cucker-Smale 2nd order model given by

(2)

xi = vi,

vi =λ

N

N∑j=1

φ(xi − xj)(vj − vi),(xi,vi) ∈ Ω× Rn

where φ(x) = φ(|x|) is a communication kernel, or influence function, originally chosen to be

(3) φ(r) =1

(1 + r2)β/2, β > 0.

The system was introduced in [33, 34] in response to the need for a model which does not require perpetuatingassumptions of connectivity of the flock to ensure unconditional alignment, a deficiency of local models.The model has seen instant success for its simplicity and amenability to analysis. Cucker-Smale protocolwith calibrated value of β = 0.4 was proposed to be used in satellite communication in Darwin mission[62]. Alignment dynamics is amenable to control problems and sustains even degenerate communications,[12, 38]. Prescribed collective outcomes can be achieved via decentralized control [9, 26]. Applications werefound to meta-heuristic optimization algorithms [18]. Flexibility of the CS model allows to incorporateindividual characteristics of agents through thermodynamic parameters [47]. Flocking behavior appearsrelevant in modeling hybrid agent systems embedded in incompressible fluid [43], in multi-scale and multi-species systems [50, 88, 63]. Other features of CS dynamics based on hierarchy, angle of vision, and emergenceof leaders are reviewed in [15].

6 ROMAN SHVYDKOY

x1 x2

xN

x1 x2

xN

D

Figure 1. Alignment, flocking, and strong flocking

3. Agent-based alignment systems

In this Chapter we study a general class of agent-based systems of Cucker-Smale type

(4)

xi = vi,

vi = λ

N∑j=1

mjφ(xi,xj)(vj − vi),(xi,vi) ∈ Ω× Rn

Here Ω denotes environment, and mj are ”masses” of agents. We always consider either open space Rn orperiodic domain Tn. The meaning of mass however depends on the context. It could be the actual physicalmass of an agent, or it could be the influence strength of that agent – its ability to effect other agents around.In this Chapter as well as for the most of our exposition to omit any forces, , focusing more on the corealignment module

We will lay out a groundwork for subsequent discussion introducing basic terminology, classification ofcommunication kernels, and presenting the most basic alignment results. We also introduce the correctormethod which allows to expand our analysis to degenerate communications, to be applied later for 3Zonemodels.

3.1. Types of communication and collective outcomes. For a given solution xi,viNi=1 to system (4)we identify the following collective outcomes of long time behavior, see Figure 1:

- alignment : limt→∞maxi |vi − v| = 0,- flocking : supi,j |xi − xj | ≤ D <∞,- strong flocking : xi − xj → xij , as t→∞.- aggregation: xi − xj → 0, as t→∞.

In fact, aggregation is not typical for forceless dynamics, and will become relevant later in Chapter 4 whenwe consider attraction forces.

Note that alignment implies strong flocking provided it occurs at a sufficiently fast rate. Indeed, if

(5)

∫ ∞0

maxi,j|vi − vj |dt <∞,

then

xi(t)− xj(t) = xi(0)− xj(0) +

∫ t

0

[vi(s)− vj(s)] ds,

hence

xij = xi(0)− xj(0) +

∫ ∞0

[vi(s)− vj(s)] ds.

This is why it is important to provide a rate of alignment whenever possible.Properties of communication kernel φ play crucial role in alignment dynamics, and may dramatically

change collective behavior of the system. We always assume that the kernels are of alignment type, i.e.

φ(x,y) ≥ 0, ∀x,y ∈ Ω.

We distinguish between the following general types of communication:

– Absolute: infx,y∈Ω φ(x,y) > 0;– Non-degenerate: φ(x,y) > 0, for all x,y ∈ Ω;– Local: there is a r0 > 0 so that φ(x,y) = 0 if |x− y| > r0;


– Symmetric: φ(x,y) = φ(y,x);– Convolution type: φ(x,y) = φ(|x− y|).

Sometimes by a local kernel we simply mean lack of global assumptions, for example, when the only infor-mation available is

(6) φ(x,y) ≥ 1, for |x− y| ≤ r0.

Majority of our examples will be of convolution type in which case we always assume that φ is sufficientlysmooth for r > 0. We say that φ is smooth, and by extension the model (4) is smooth, if φ ∈ C2(Rn), i.e.φ is regular at the origin also. Otherwise, we say that the kernel and the model are singular. The mostimportant example of a singular kernel for us will be the power kernel

(7) φ(r) =h(r)

rβ,

where h is a possible cut-off function if we consider local kernels. If β = n + α, 0 < α < 2, then thisbecomes the kernel of a localized fractional Laplacian, which we will study in detail in Chapters 8 and 9.The strength of communication, with or without any presumed structure of the kernel, can be expressed interms of integrability condition either at the close range or long range

Long range (“heavy tail”):

∫ ∞r0

φ(r) dr =∞,(8)

Short range:

∫ r0

0

φ(r) dr =∞.(9)

Two special examples of non-convolution kernels will play prominent roles in our discussion. First is anon-symmetric communication protocol introduced by Motsch and Tadmor [71]:

(10) ψ(xi,xj) =φ(|xi − xj |)∑kmkφ(|xi − xk|)

.

This averaging,∑jmjψ(xi,xj) = 1, allows to avoid deficiencies of uniform CS averaging associated with far

from equilibrium flock configurations, see Section 3.8. Both singular and Motsch-Tadmor kernels are meantto emphasize local interactions over global ones whenever such communication is deemed more realistic.

So far all presented examples belong to a class of so called metric kernels, meaning that communicationdepends on the Euclidean distance between agents. In some biological systems, such as flocks of starlings,communication follows somewhat different protocol where the strength of interaction depends on the densityof crowd between communicating agents. When distances are measured in terms of mass, the kernel andmodel are called topological. The example that we will discuss in detail in Section 7.4 is given by

(11) φij(x) =1

dτijψ(|xi − xj |), dij =

∑k:xk∈Ωij

mk

1n

,

where ψ is a metric component and dnij would constitute the mass of a given communication domain Ωij . Ifthe domain is symmetric Ωij = Ωji, then so is the kernel.

3.2. Momentum, Energy, and Maximum Principle. System (4) with symmetric kernels, as opposedto non-symmetric, conserves the total momentum

(12) v =1

M

∑i

mivi, M =

N∑i=1

mi,d

dtv = 0.

Due to conservation of momentum, the center of mass moves with a constant velocity

(13) x =1

M

∑i

mixi,d

dtx = v.

For convolution type kernels, conservation of momentum can be used to shift the reference frame centeredat x, due to Galilean invariance of the system:

(14) xi → xi − tv, vi → vi − v.

8 ROMAN SHVYDKOY

So, in this case we can assume without loss of generality that v = 0. In general, such translation invarianceis not available. Nonetheless, if alignment occurs, then necessarily all vi → v. In other words, we candetermine the limiting velocity from initial conditions.

Let us consider the following variation and dissipation functions:

V2 =1

2

∑i,j

mimj |vi − vj |2

I2 =1

2

∑i,j

mimjφ(xi,xj)|vi − vj |2.(15)

The system has the classical kinetic energy as well defined by

E =1

2

N∑i=1

mi|vi|2.

In case if v = 0, then V2 = 2ME , however it is not prudent to use energy as a measure of alignment E in thenon-symmetric case, simply because we do not know if 0 would remain to be the momentum of the systemfor all time.

The following energy law is easily verified:

(16)d

dtV2 = −2λMI2.

At this point one can obtain an `2-base alignment result under absolute communication: if inf φ = c0 > 0,then I2 ≥ c0V2, and hence

V2 ≤ −2c0λMV2.

Hence,

V2(t) ≤ V2(0)e−2c0λMt.

This result provides exponential alignment ”on average”, specifically L2-average, which does not translatewell into individual information on agents. Indeed, one only obtains

(17) |vi − vj | ≤1

mimjV2(0)e−δt.

This estimate clearly deteriorates in the large crowd limit N → ∞ when all masses vanish mi → 0. Inorder to improve upon (17) we must resort to an `∞-based argument and use the maximum principle to bediscussed later in Section 3.4.

3.3. Connectivity and spectral method. In the last section we saw that flocking behavior holds triviallyunder global communication condition. At the same time if communication is local it is easy to produce anexample of two agents or disconnected flocks separated by a distance longer than communication range, thatwould scatter in opposite directions and not align, see the Figure 2. So, it is clear that ultimately connectivityis the key to flocking behavior. Let us make this more precise. We say that the flock is r0-connected if for

any pair of agents xi and xj there exists a chain of agents xkpPijp=1 with end-points at xi and xj and such

that all |xkp − xkp+1| < r0. If the flock is r0-connected within a communication range, λφ(r0) = ε > 0, then

alignment can be recovered as follows. Note that the shortest chain connecting any pair of agents has norepeated agents in the chain. Hence, every chain is limited to length N . We can then estimate V2, assumingfor simplicity that all agents have the same mass mi = 1

N ,

V2 =1

2N2

∑i 6=j|vi − vj |2 ≤

1

2N2

∑i 6=j

Pij∑p=1

|vkp − vkp+1|2

≤ λ

2εN2

∑i 6=j

Pij∑p=1

φ(|xkp − xkp+1|)|vkp − vkp+1

|2

≤ N(N − 1)

2εN2λ∑k′ 6=k′′

φ(|xk′ − xk′′ |)|vk′ − vk′′ |2 . N2I2.


The desired differential inequality follows

V2 ≤ −1

N2V2.

The argument produces a bad dependence on N , and as such is not suitable in the limit N → ∞. Acontinuous analogue of connectivity condition requires more elaboration and is discussed in [70].

Connectivity at all times is hard to verify of course. However, it is guaranteed to hold provided theinitial configuration is connected and the communication strength λ is large enough. This is because one canensure in this case that the system aligns almost instantaneously before any disconnection becomes possible.Indeed, suppose initially the system is r0/2 connected, and λφ(r0) is large. Then for some large Λ > 0 wewill have

d

dtV2 ≤ −ΛV2,

for as long as the system is r0-connected. So, if a pair xi, xj is initially at most r0/2 apart, then

|xi(t)− xj(t)| ≤ |xi(0)− xj(0)|+ C

Λ.

This shows that the same pair will never get r0-disconnected and hence the connectivity is preserved at alltimes.

If the kernel has a long range tail, r0 = ∞, then obviously every flock is r0-connected. But how strongthat long range communication needs to be in order to actually ensure flocking? Let us illustrate the answerby the following example.

Example 3.1. Let the kernel be φ(r) = 1rβ

for r > r0, for simplicity, and let x = x1 = −x2 > r0 andv = v1 = −v2 > 0. This symmetry is preserved in time. Then the system (4) becomes

dx

2β−1xβ+

dv

1= 0.

This equation admits a conservation law

J = v +1

2β−1(1− β)xβ−1.

If β > 1, and the initial velocity is large enough, then J(t) = J0 > 0, and hence v(t) ≥ J0 holds true for alltimes. This creates permanent misalignment between the two velocities at hand: v and −v.

So, we can see that integrability of the kernel tail is the key. To obtain a general result Cucker and Smaleproposed to quantify connectivity strength of the flock as it evolves and prove that that strength is sustainedby a non-integrable tail. This lies at the heart of their spectral method which we discuss next.

Let us consider the matrix A = aij(t)Ni,j=1 ⊗ In×n, where aij = λφ(xi(t),xj(t)), i 6= j, and aij = 0, ifi = j. Note that aij 6= 0 if and only if the corresponding agents and “connected” in the sense that theycommunication through the influence kernel φ. By analogy with the graph theory we call it the adjacencymatrix of the flock. In fact, one can consider the actual adjacency matrix associated with the flock underthe above connectivity definition: A = aij(t)Ni,j=1 ⊗ In×n, where aij = 1 if aij 6= 0, and 0 otherwise. Let

D = diagb1, . . . , bN⊗ In×n, where bi =∑j aij , and D = diagb1, . . . , bN⊗ In×n, where bi =

∑j aij . Note

that each bi is precisely the degree of the vertex xi, i.e. the number of other agents to which it is connected.In this notation the system (4) can now be written in terms of the grand velocity vector V = (v1, . . . ,vN ),

and the Laplacian associated to A, L = D −A, namely,

(18)d

dtV = −LV.

The matrix L is non-negative definite, and hence the spectrum consists of a sequence 0 = κ1 ≤ κ2 ≤ ... ≤ κN .The average grand vector V = (v, . . . , v) is obviously a member of the kernel of L, hence κ1 = 0. The nexteigenvalue κ2 is called the Fiedler number, although the classical Fiedler number is one associated to theLaplacian L = D − A is a similar way, we denote it κ2. By the min-max theorem κ2 is given by

(19) κ2 = min∑i vi=0

〈LV,V〉|V|2 .

10 ROMAN SHVYDKOY

bb

b

b

b

b

b

b

b

b

xj

xi

r0

r0

r0

r0

> r0

Figure 2. Connected and disconnected flocks

A simple fact to verify is that κ2 6= 0 if and only if the graph is connected, and the relationship to κ2 isgiven by (see [34, Proposition 2]):

(20) κ2 ≥ κ2 mini,j:aij 6=0

aij .

For this reason, we can call κ2 a weighted Fiedler number, which captures not only algebraic connectivityof the flock as a graph, but also the collective strength of the connection weighted by the kernel φ. Withthe use of the weighted Fiedler number κ2 = κ2(t), which, let’s recall, depends on time, we can measurealignment in (18) by writing the energy law as

d

dt|V − V|22 = −2〈L(V − V), (V − V)〉 ≤ −2κ2(t)|V − V|22.

Consequently,

|V(t)− V|2 ≤ |V0 − V|2 exp

−∫ t

0

κ2(s) ds

.

We can see that the divergence of the integral inside leads to alignment, which can be seen as a quantitativemeasure of connectivity as a function of time. Let us state it precisely.

Lemma 3.2. If∫∞

0κ2(s) ds =∞, then the flock aligns.

At the center of the original result of Cucker and Smale was a statement that for kernels of type (3)the integral of the weighted Fiedler number is indeed divergent. Here, clearly, the flock remains alwaysalgebraically connected , thus κ2 = N , and one can appeal directly to (20) to restate the problem in termsof control on the decay of the adjacency matrix. The original theorem of Cucker and Smale [33, 34] is thefollowing.

Theorem 3.3 (Cucker-Smale). Let φ(r) = 1(1+r2)β/2 . Then every solution aligns exponentially and flocks

strongly for β ≤ 1, and conditionally if β > 1.

In Section 3.4 this result will be proved using a more Lyapunov function approach, which paves the wayto extensions into kinetic and hydrodynamic systems. We will also state a sharp condition that ensuresalignment even for no heavy tail case. Let us not, however, underestimate the spectral method as in someoccasions it is the only one available if no specific structural information is known about the kernel.

3.4. Alignment with heavy tail communication. The main goal in this section will be to prove theCucker-Smale Theorem 3.3 in more general settings of convolution type heavy tail kernel using a methodbased on the maximum principle. We will see that the argument is rather versatile and is easily adaptableto other non-symmetric kernels. So, we consider the classical Cucker-Smale system

(21)

xi = vi,

vi = λ

N∑k=1

mkφ(xi − xk)(vk − vi),(xi,vi) ∈ Rn × Rn.

We assume that φ is monotonely decreasing and everywhere positive. It will be useful to use sometimes thefollowing shortcut notation:

(22) xij = xi − xj , vij = vi − vj , φij = φ(xi − xj), etc.


Let us also consider the amplitude and flock diameter:

(23) D = maxi,j|xij |, A = max

i,j|vij |.

Theorem 3.4. Suppose φ is decreasing, positive and satisfying with initial condition D0,A0

(24)

∫ ∞D0

φ(r) dr >A0

λM.

Then the solution aligns and flocks exponentially fast:

(25) supt≥0D(t) ≤ D, A(t) ≤ A0e

−tλMφ(D).

In particular, every solution flocks provided the kernel satisfies the heavy tail condition (8).

To make the proof perfectly rigorous, let us recall the classical Redemacher Lemma, which we will usethroughout.

Suppose f(x, t) : X × R+ → R is a Lipschitz in time function uniformly in x, where X is an arbitraryindex set. So, ∃L > 0 such that for all t, s, x we have

|f(x, t)− f(x, s)| ≤ L|t− s|.Suppose that at any time t there is a point x(t) ∈ X such that

f(x(t), t) = supx∈X

f(x, t) := F (t).

Note that F (t) is a Lipschitz function with the same constant L. Indeed, let t, s ∈ R+ and S(t) > S(s).Then

F (t)− F (s) = f(x(t), t)− f(x(t), s) + f(x(t), s)− f(x(s), s)︸︷︷︸≤0

≤ f(x(t), t)− f(x(t), s) ≤ L|t− s|.

Consequently, F is absolutely continuous on any finite interval, i.e.

F (t)− F (s) =

∫ t

s

m(τ)dτ,

where ‖m‖∞ ≤ L∞loc, and hence F ′ = m a.e.

Lemma 3.5. If f(x, ·) is differentiable everywhere in t for all x ∈ X, then F ′(t) = ∂tf(x(t), t) holds at anypoint where F ′ exists.

Indeed computing one-sided derivative from the right we have

F ′(t) = limh→0+

f(x(t+ h), t+ h)− f(x(t), t+ h) + f(x(t), t+ h)− f(x(t), t)

h

≥ limh→0+

f(x(t), t+ h)− f(x(t), t)

h= ∂tf(x(t), t).

Taking h < 0 proves the opposite inequality.

Proof of Theorem 3.4. Let us represent A as

A = max|`|=1,i,j

`(vij).

Note that the maximum is takes over a fixed compact set not changing in time. So, using Rademacher’slemma we can pick ` and i, j at each time t for which the maximum is achieved. Using the velocity equationwe obtain

(26)d

dt`(vij) = λ

N∑k=1

mkφik`(vki) +mkφjk`(vjk)

12 ROMAN SHVYDKOY

Now notice that `(vki) ≤ 0 and `(vjk) ≤ 0. One can see it by adding and subtracting vj in the first case andvi in the second and using maximality of `(vij). So, then we can pull out the minimal value of the kernel inboth sums:

d

dt`(vij) ≤ −λφ(D)

N∑k=1

mk`(vij) = −λMφ(D)`(vij).

So, we obtain

(27)

d

dtA ≤ −λMφ(D)A

d

dtD ≤ A.

This system of ordinary differential inequalities (ODIs) has a decreasing Lyapunov function given by L =

A+ λM∫ D

0φ(r) dr. This, in particular, implies that

λM

∫ D(t)

0

φ(r) dr ≤ A0 + λM

∫ D0

0

φ(r) dr, ∀t > 0.

Consequently, D(t) ≤ D, where D is obtained from the equation

(28) λM

∫ DD0

φ(r) dr = A0,

which is guaranteed to have a finite solution due to (24). Then, A ≤ −λMφ(D)A and the theorem follows.

Solving equation (28) allows one to provide explicit decay rates for solutions of (27) for some kernels. Inparticular, for the classical Cucker-Smale kernel

φ(r) =1

(1 + r2)β2

,

one obtains

D ≤([

1− βλM

A0 + (1 +D20)

1−β2

] 21−β

− 1

) 12

, β < 1,

rate =λM[

1−βλM A0 + (1 +D2

0)1−β

2

] β1−β

.

(29)

(here one replaces φ with a smaller but explicitly integrable kernel r

(1+r2)β+1

2

), and

(30) D ≤(e

2λMA0(1 +D2

0)− 1) 1

2

, rate =λM

eA0λM (1 +D2

0)12

, β = 1.

The proof of Theorem 3.4 is rather flexible and can be adapted to other models, including those withnon-symmetric communication, see Section 3.8.

A notable byproduct of the arguments presented above is that |·| can be any norm defined on Rn, with unitfunctionals |`| = 1 chosen in the dual norm. Even though all norms on Rn are equivalent, their equivalencyconstants can be large. However, the basic system of inequalities (27) would still hold true with exact samecoefficients, independent of chosen norm.

3.5. Stability. Suppose now we have two close initial conditions for the same flock with the same momen-tum:

V = (vi)Ni=1, X = (xi)

Ni=1, V = (vi)

Ni=1, X = (xi)

Ni=1,

and v = ¯v. We show that if initially these parameters are close, then they will remain close uniformly forall time. To formulate this precisely let us introduce the measure of distance between flocks:

(31) ‖X− X‖ = maxi|xi − xi|, ‖V − V‖ = max

i|vi − vi|.


Here | · | denotes any norm on Rn. Much in the spirit of the previous proofs let us derive a system of ODIsfor these two quantities. First, we clearly have

d

dt‖X− X‖ ≤ ‖V − V‖.

Second, let us represent

‖V − V‖ = maxi,|`|=1

`(vi − vi),

and by Rademacher’s lemma apply derivative in time with a maximizing pair i, `:

d

dt`(vi − vi) = λ

N∑k=1

mkφ(xik)`(vki)−mkφ(xik)`(vki)

= λ

N∑k=1

mk(φ(xik)− φ(xik))`(vki) + λ

N∑k=1

mkφ(xik)[`(vk − vk)− `(vi − vi)]

≤ 2Mλ|∇φ|∞A‖X− X‖+ λφ(D)

N∑k=1

mk[`(vk − vk)− `(vi − vi)]

= 2Mλ|∇φ|∞A0e−tλMφ(D)‖X− X‖ −Mλφ(D)‖V − V‖

Let us recall that the bounds on the diameters ultimately depend on the initial conditions via (28). If

the initial flock X,V is known and the perturbation X, V is relatively small, then those diameters can bequantified by initial values of D0 and A0. So, let us simply denote

γ = minφ(D), φ(D).We have obtained the system

(32)

d

dt‖V − V‖ ≤ 2Mλ|∇φ|∞A0e

−tλγM‖X− X‖ − λγM‖V − V‖d

dt‖X− X‖ ≤ ‖V − V‖.

Let us simply rewrite it as

(33) x′ ≤ v, v′ ≤ ae−btx− bv.It is elementary to obtain a bound on solutions. Indeed, denoting w = vebt we obtain

x′ ≤ we−bt, w′ ≤ ax.Multiplying by factors to equalize the right hand sides, we obtain

d

dt(ax2 + e−btw2) ≤ 4axwe−bt ≤ 2e−bt/2

√a(ax2 + e−btw2).

This immediately implies

ax2 + ebtv2 ≤ 4√a

b(ax2

0 + v20).

We can read off bounds for each parameter individually:

(34) x ≤ 2

a1/4b1/2

√ax2

0 + v20 , v ≤ e−bt/2 2a1/4

b1/2

√ax2

0 + v20 .

Theorem 3.6. The following bound holds for any pair of solutions to (21) with the same momentum:

(35) a‖X− X‖2 + ebt‖V − V‖2 ≤ 4√a

b(a‖X0 − X0‖2 + ‖V0 − V0‖2),

where a = 2Mλ|∇φ|∞A0 and b = λM minφ(D), φ(D).

14 ROMAN SHVYDKOY

3.6. Singular kernels and the issue of collisions. Let us introduce into consideration the singular kernels(7), which will play a crucial role in macroscopic description of the system. Clearly, singularity at the originemphasizes predominantly local communication as is natural in some applications. However, with singularitywe stumble upon the issue of well-posedness of the system (4) in the case when agents encounter collisions.In fact collisions are common in the bounded kernel case.

Example 3.7. Let us assume that φ = 1 in a neighborhood of 0. Let us arrange two agents x = x1 = −x2 with0 < x(0) = ε 1. And let v1 = −v2 < 0 be very large. Clearly x(t) will remain in the same neighborhoodof 0 as where it has started, and so the system reads

d

dtx = v,

d

dtv = −2v.

Solving it explicitly we can see that the two agents will collide at the origin.

Heuristically, however, strong singularity should prevent such collisions. The induced alignment forcesshould become strong enough to correct velocities of converging agents before collision happens in the firstplace. Exactly how singular the kernel should be can be seen from the following example.

Example 3.8. Let the kernel be given by (7) and let us consider the same setup as previously. Then weobtain the system

d

dtx = v,

d

dtv = −2

v

xβ.

This system has a conservation law provided β < 1: v+ 2x1−β

1−β = C0. So, if initially C0 0, then v < C0 0

as well. This means that x will reach the origin in finite time.

This example demonstrates that the threshold singularity necessary to prevent collisions must be non-integrable. It is indeed true as we prove in the following theorem.

Theorem 3.9. Under the strong singularity condition (9) the flock experiences no collisions between agentsfor any non-collisional initial datum. Consequently, any non-collisional initial datum gives rise to a uniqueglobal solution.

In view of global existence and absence of collisions the content of Theorem 3.4 holds true as statedprovided the kernel is singular (9) condition.

Proof. For a given non-collisional initial data (xi,vi)i let us assume that collision occurs first time at t = T ∗.Denote by I∗ ⊂ 1, ..., N the indexes of agents that collided at one point in space – note that this may notbe unique collection. Consequently, there is a δ > 0 such that |xik(t)| ≥ δ for all i ∈ I∗ and k ∈ Ω\I∗. Letus denote

D∗(t) = maxi,j∈I∗

|xij(t)|, A∗(t) = maxi,j∈I∗

|vij(t)|, t < T ∗.

Directly from the characteristic equation we obtain ddt|D∗| ≤ A∗, and hence

(36) − d

dtD∗ ≤ A∗.

For velocity variation we obtain, using a maximizing triple ` ∈ (Rn)∗, i, j ∈ I∗:

d

dtA∗ =

1

N

N∑k=1

φik`(vki)− φkj`(vkj) =1

N

∑k∈I∗

φik`(vkj − vij) + φkj`(−vki − vij)

+1

N

∑k 6∈I∗

φik`(vki)− φkj`(vkj).

In the first sum we notice that all terms are negative, so we can pull out the minimal value of the kernel,which is φ(D∗). In the second sum, all the distance |xik|, |xjk| stay away from zero up to T ∗. So, the kernelsand the whole sum remains bounded. In summary we obtain

d

dtA∗ ≤ C1 − C2φ(D∗)A∗.


Considering the energy functional

E(t) = A∗(t) + C2

∫ 1

D∗(t)φ(r) dr,

we readily find that ddtE ≤ C1, hence E remains bounded up to the critical time. This means that D∗(t)

cannot approach zero value.The global existence part is now a routine application of the Picard iteration and the standard continuation

argument.

When the power of kernel β ≥ 2 a quantitative estimate on the minimal distance between agents can bederived. Assuming that the singularity assumption holds the local region r < r0:

(37) φ(r) &1r<r0rβ

,

let us consider the collision functional

(38) C =

1

N2

N∑i,j=1

1

(|xij | ∧ r0)β−2, β > 2

1

N2

N∑i,j=1

ln(|xij | ∧ r0), β = 2.

For β > 2 we have for the derivative

dCdt

=(2− β)

N2

N∑i,j=1

ddt

(|xij | ∧ r0

)(|xij | ∧ r0)β−1

≤ |β − 2|N2

N∑i,j=1

1

(|xij | ∧ r0)β−1|vij |1|xij |<r0

≤ |β − 2|

1

N2

N∑i,j=1

v2ij

1

|xij |β1|xij |<r0

1/2 1

N2

N∑i,j=1

|xij |2−β 1|xij |<r0

1/2

≤ C√I2

√C.

This implies

(39)√C(t) ≤

√C(0) + C

∫ t

0

√I2(s) ds,

and recalling that I2 is integrable on R+ we conclude

(40) C(t) . t.For β = 2, a similar computation gives d

dtC ≤ C

√I2, hence C(t) .√t. We thus arrive at the following

bounds

(41) |xij(t)| ≥

c

t1

β−2

, β > 2,

ce−C√t, β = 2.

3.7. Degenerate communication. Corrector Method. Non-degenerate communication is not alwaysrealistic if we think of biological systems such as flocks. Indeed, every bird has only limited sensory abilities.In addition, in close range alignment force may not even be the most dominant as it appears for examplein 3Zone models which we will discuss in Chapter 4. The case when the kernel is degenerate, i.e. vanishesat some points, presents obvious difficulty with implementation of the Lyapunov function approach weconsidered so far. The main difficulty is of course lack of any coercivity in the energy law

(42)d

dtV2 = −2I2.

If the kernel is non-degenerate only in a bounded range r ≤ r0, for example as depicted in Figure 3, thenthere still exists a dynamical mechanism to restore communication and as a result alignment. Indeed, iftwo agents xi, xj are not yet aligned, then their velocities are not the same. Hence, they would tend to

16 ROMAN SHVYDKOY

b

φ(r)

r0

AlignmentIndifference

Figure 3. Communication degenerate in close range

separate from each other, and provided no other influence changes their course, they eventually reach thecommunication range r > r0. In this section we present a method introduced in [38] which allows to partiallyfix the lack of coercivity by introducing a proper energy corrector, hence the corrector method. The methoduses higher order fluctuation functionals given by

Vp =1

pN2

N∑i,j=1

|vi − vj |p, p ≥ 1,

Ip =1

pN2

N∑i,j=1

φ(xi,xj)|vi − vj |p.(43)

We observe that Vp’s are non-increasing. Indeed,

d

dtVp =

1

N3

∑i,j,k

|vij |p−2vij · (vkiφki − vkjφkj) =2

N3

∑i,j,k

|vij |p−2vij · vkiφki

=1

N3

∑i,j,k

(|vij |p−2vij − |vkj |p−2vkj) · vkiφki

=1

N3

∑i,j,k

(|vij |p−2vij − |vkj |p−2vkj) · (vkj − vij)φki,

with the convention that |vij |p−2vij = 0 if i = j. The right hand side is non-positive due to the elementaryinequality (

|a|p−2a− |b|p−2b)· (a− b) ≥ 0.

The two special cases, i = j and k = j, produce the term −|vik|p − |vij |p. So, in general we have anN -dependent inequality

(44)d

dtVp ≤ −

p

NIp, p ≥ 1.

For the case p = 2 the energy law is of course N -independent, (42).

Theorem 3.10. Suppose the kernel φ ≥ 0 is either smooth or satisfying (9). Suppose also that it dominatesa monotone heavy tail: there exists a non-increasing Φ(r), such that for some r0 > 0

(45) φ(r) ≥ Φ(r), ∀r > r0, and

∫ ∞r0

Φ(r) dr =∞.

Then

(i) Any solution to the discrete system (4) aligns: V2(t) ≤ CNt , and D(t) ≤ DN , with constants depending

on N .(ii) If φ is smooth, then any solution to the discrete system (4) aligns: V4 → 0, with a rate independent

of N .


The advantage of (i) over (ii) is that it gives a faster, although N -dependent, rate as well as flocking. Italso holds for singular kernels satisfying non-collision condition (9). However, it is not extendable to themacroscopic or kinetic case, while (ii) is.

Proof of Theorem 3.10 (i). We start by defining the following corrector

G =1

N2

N∑i,j=1

|vij |ψ(dij)χ(|xij |),

where dij is a longitudinal displacement function defined by

(46) dij = −xij ·vij|vij |

,

and the two auxiliary functions χ : R+ 7→ R+ and ψ : R 7→ R+ are defined by

χ(r) =

1, r < r0,

2− rr0, r0 ≤ r ≤ 2r0,

0, r > 2r0

and ψ(d) =

0, d < −r0,

d+ r0, |d| ≤ r0,

2r0, d > r0.

Let us compute the derivative of the corrector

(47)d

dtG = − 1

N2

N∑i,j=1

|vij |21|dij |<r0χ(|xij |) +R1 +R2 +R3,

where

R1 =2

N3

N∑i,j,k=1

vij|vij |

· vkiφ(xik)ψ(dij)χ(|xij |),

R2 = − 2

N3

N∑i,j,k=1

xij ·(I− vij ⊗ vij

|vij |2)

vkiφ(xik)1|dij |<r0χ(|xij |)

R3 =1

N2

N∑i,j=1

|vij |ψ(dij)χ′(|xij |)

xij|xij |

· vij .

The first term on the right hand side of (47) patches up communication where it was originally missing –at the close range. Let us address it first. Without loss of generality we can assume that Φ is a boundeddecreasing function on R+. Hence,

1r<r0 + φ(r) ≥ cΦ(r),

for all r > 0 and some c > 0. Using 1|dij |<r0 ≥ 1|xij |<r0 , we have

− 1

N2

∑i,j

|vij |21|dij |<r0χ(xij) ≤ −1

N2

∑i,j

|vij |21|xij |<r0

= − 1

N2

∑i,j

|vij |2(1|xij |<r0 + φ(xij)) +1

N2

∑i,j

|vij |2φ(xij)

≤ −cΦ(D)V2 + I2.

Proceeding to the error terms, by construction of the auxiliary functions, we have

|R1|, |R2| . I1,

and

R3 ≤1

N2

∑i,j

|vij |2|χ′(xij)| ≤1

N2

∑i,j

|vij |21r0<|xij |<2r0 .1

N2

∑i,j

|vij |2φ(xij) = I2.

Hence, with constants a, b, c only depending on φ, we obtain

d

dtG ≤ −cΦ(D)V2 + aI2 + bI1.

18 ROMAN SHVYDKOY

We can now define a new Lyapunov functional:

(48) L = G + aV2 + bNV1,d

dtL ≤ −cΦ(D)V2.

Using that ddtD ≤ CN

√V2, we find another Lyapunov functional:

L = L+c

CN

√V2

∫ D0

Φ(r) dr.

Consequently, due to heavy tail on Φ, we have flocking D(t) ≤ DN for all time and for some N -dependentDN . Returning to (48) we conclude that

A :=

∫ ∞0

V2(t) dt <∞.

So, on every time interval [T, eAT ] there is a t such that V2(t) ≤ 1t . By monotonicity, this implies a similar

bound for all t.

Proof of Theorem 3.10 (ii). To achieve N -independent, although slower, rate we consider a third order cor-rector given by

G3 =1

N2

∑i,j

|vij |3ψ(dij)χ(|xij |),

with ψ and χ defined previously. In this case

d

dtG3 = − 1

N2

N∑i,j=1

|vij |41|dij |<r0χ(|xij |) +R1 +R2 +R3,

with

R1 =6

N3

N∑i,j,k=1

|vij |vij · vki φik ψ(dij)χ(|xij |),

R2 =−2

N3

N∑i,j,k=1

|vij |2 xij ·(I− vij ⊗ vij

|vij |2)

vkiφki 1|dij |<r0χ(|xij |)

R3 =1

N2

N∑i,j=1

|vij |3ψ(dij)χ′(|xij |)

xij|xij |

· vij .

The gain term is estimated as before with the use of a priori uniform bound on velocities |vi(t)| ≤ |v(0)|∞:

(49) − 1

N2

N∑i,j=1

|vij |41|dij |<r0χ(|xij |) ≤ −cΦ(D)V4 + |v(0)|2∞I2.

Next,

R3 .1

N2

∑i,j

|vij |3φ(|xij |) . I2.

By Young’s inequality for a small ε > 0 to be specified later,

R2 .ε

N2

N∑i,j=1

|vij |4|xij |2χ(|xij |)2 +|φ|∞ε

1

N2

N∑i,k=1

|vki|2φki.

Note that χ(|xij |) = 0 if |xij | > 2r0. So,

R2 .ε

N2

N∑i,j=1

|vij |41|xij |≤2r0 +1

εN2

N∑i,k=1

|vki|2φki

≤ ε

N2

N∑i,j=1

|vij |41|xij |≤r0 +ε

N2

N∑i,j=1

|vij |41r0<|xij |<2r0 +1

εN2

N∑i,k=1

|vki|2φki


For ε sufficiently small the first sum gets absorbed into the gain term (49). The second and third sums aredominated by I2. The term R1 can be estimated in exact same manner. We obtain

(50)d

dtG3 ≤ −cΦ(D)V4 + aI2.

By analogy with previous proof we define the Lyapunov functional L = G3 + aV2 and conclude that

(51)

∫ ∞0

Φ(D(t))V4(t) dt <∞,

with the bound being independent of N . Furthermore, due to uniform bound on velocity, D(t) ≤ ct + D0.Thus,

(52)

∫ ∞0

Φ(ct+D0)V4(t) dt <∞.

By virtue of the heavy tail condition on Φ, V4 cannot be bounded away from the zero. And in view of itsmonotonicity, V4 → 0.

Remark 3.11. In either of the results above we can actually extract a specific rate of alignment if we makean explicit assumption about the tail of the kernel. Thus, if

φ(r) ∼ 1

rβ, ∀r > r0,

and β ≤ 1, then from (51), ∫ ∞1

1

tβV4(t) dt <∞,

where V is the corresponding functional. So, if β = 1, then there exists an A > 0 such that for any T > 1there exists a t ∈ [T, TA] such that V4(t) < 1

ln t . Since ln t is proportional to lnT for all t ∈ [T, TA] thisproves the above bound for all large times. If β < 1, then we argue that for some large A > 0 and all T > 0we find t ∈ [T,AT ] such that V4(t) ≤ 1

t1−β. But the latter is comparable for all values of t ∈ [T,AT ]. Thus

we obtain the power rate as above for all t.Let us summarize the obtained results.

V4(t) .1

ln t, β = 1,

V4(t) .1

t1−β, β < 1.

(53)

Of course the same argument applies to V2 under the conditions of part (ii).

3.8. Multi-flocks. Clusters. Multi-species. In the case when a flock is geometrically non-homogeneousthe classical Cucker-Smale system may yield a non-realistic dynamics. Let us consider one such example.Suppose all the masses are the same mi = 1

N , and suppose that we have two well separated clusters in theflock, one containing a large collection of agents N1, and one is small N2 N1, see Figure 4. Since massesrepresent influence strengths the agents, one would expect that dynamics of the small flock will be largelyindependent of the large flock, and so its agent masses should rather be 1

N2. This is, however, contrary to

what CS system would show: for any i in the small flock, since φ(xi−xj) ∼ 0 for all j in the large flock, wehave

vi ∼1

N1 +N2

N1∑j=1

φ(xi − xj)(vj − vi) ∼ 0.

So, the dynamics of the small flock is stalled according to the classical CS description. To remedy thesituation, Motsch and Tadmor in [71] proposed to consider a model where uniform averaging as in CS isreplaced with adaptive averaging

vi =1∑

k φ(|xi − xk|)N∑k=1

φ(xi − xk)(vk − vi).

When applied to the described situation, we will have for each i in the small flock∑k

φ(|xi − xk|) ∼ N2.

20 ROMAN SHVYDKOY

Figure 4. Large flock overtakes dynamics of a small flock

As a result the model rebalances masses to the correct value 1N2

.More generally, one can consider the model with non-homogeneous mass distribution

(54) vi =λ∑

kmkφ(|xi − xk|)N∑k=1

mkφ(xi − xk)(vk − vi).

Despite the fact that the kernel is no longer symmetric in the Motsch-Tadmor model, one can still obtainthe full analogue of Theorem 3.4:

(55)

∫ ∞D0

φ(r) dr >A0|φ|∞

λ⇒ A(t) ≤ A0e

−tλ|φ|−1∞ φ(D).

Notice that the mass is no longer present in the formula for the rate of convergence since it is simply scaledout of the new communication protocol. Indeed, arguing as before, we obtain

d

dt`(vij) = λ

N∑k=1

mkφik∑Np=1mpφip

`(vki) +mkφjk∑Np=1mpφjp

`(vjk)

≤ λ

M |φ|∞φ(D)

N∑k=1

mk[`(vki) + `(vjk)] = − λ

|φ|∞φ(D)`(vij).

(56)

The rest of the proof follows that of Theorem 3.4.Due to lack of symmetry the limiting velocity of the flock is no longer determined by initial condition,

but rather becomes an emergent quantity of the dynamics. Since all the differences vij vanish exponentiallyfast, it implies that velocities do in fact converge to a time independent limit as seen from integrating thevelocity equation:

limt→∞

vi(t) = vi(0) +

∫ ∞0

λ

N∑k=1

mkφik∑Np=1mpφip

vki(s) ds.

Let us recognize still one more issue of the alignment formula (55) – the rate depends on the global boundon the diameter of the full system D. While global alignment may as well be slow for separated flocks thealignment within each flock may happen at a much faster rate proportional to φ(Di). In order to capturethis phenomenon we need to incorporate multi-scale communication into the model. Let us assume that wehave A well separated flocks indexed by α = 1, . . . , A. We have velocities vαi and positions xαi within eachflock indexed by i = 1, . . . , Nα. When α 6= β the communication between flocks can be approximated by

φ(xαi − xβj) ∼ φ(Xα −Xβ).

where Xα is the center of mass, Xα = 1Mα

∑Nαi=1mαixαi. So,

vαi = λ

Nα∑j=1

mαjφ(xαi − xαj)(vαj − vαi) + ε∑β 6=α

Mβφ(Xα −Xβ)(Vβ − vαi),

where Vα = 1Mα

∑Nαi=1mαivαi, and ε λ. To make the model more flexible we may assume that commu-

nications within flocks may be different and also distinct from inter-flock communication. We thus consider


the following model

(57)

xαi = vαi,

vαi = λα

Nα∑j=1

mαjφα(xαi − xαj)(vαj − vαi) + ε

A∑β=1β 6=α

MβΨ(Xα −Xβ)(Vβ − vαi),

It follows from the construction that the macroscopic variables Xα,Vα satisfy the upscaled system

(58)

Xα = Vα,

Vα = ε∑β 6=α

MβΨ(Xα −Xβ)(Vβ −Vα).

To study collective behavior of the multi-flock system (57) let us consider the following size parameters

Dα = maxi,j|xαi − xαj |, D = max

α,β|Xα −Xβ |

Aα = maxi,j|vαi − vαj |, A = max

α,β|Vα −Vβ |.

The alignment of macroscopic quantities follows from the same system of ODIs as we derived in the classicalcase:

(59)

A ≤ −εMΨ(D)AD ≤ A.

Thus, under heavy tail condition on Ψ the consensus directions Vα will in fact align exponentially fastaccording to Theorem 3.4. To understand the alignment within each flock we consider a maximizing triple`, i, j for an α-flock and apply computation (26):

d

dt`(vαi − vαj) ≤ −λαMαφ(Dα)Aα − εMΨ(D)Aα.

we obtain the system of ODIs:

(60)

Aα ≤ −λαMαφ(Dα)Aα − εMΨ(D)AαDα ≤ AαA ≤ −εMΨ(D)AD ≤ A

Ignoring −εMΨ(D)Aα in the Aα equation for the moment, we see that the α-flock decouples from the rest.Consequently, a fast internal alignment insues applying Theorem 3.4.

Theorem 3.12 (Fast internal flocking). If for a given α ∈ 1, . . . , A the kernel φα has a heavy tail, theα-flock aligns exponentially fast at a rate functionally dependent on φα, initial data, and λα:

maxi|vαi(t)−Vα(t)| . e−δt.

It is interesting to note that this alignment process is completely independent from the inter-flock com-munication. So, long range internal communication leads to local emergence despite potentially destabilizinginfluence of the outside crowd. On the other hand, if the inter-flock communication Ψ is global, e.g. satisfiesthe heavy tail condition (8), then the global alignment occurs even if internal communications are weak oreven absent. This is clear from (60) if we drop −λαMαφ(Dα)Vα and conclude boundedness of D from thelast two equations. Alignment rate in this case is global but occurs on the slow time scale.

Theorem 3.13 (Slow global flocking). Assuming that Ψ has a heavy tail and all φα ≥ 0, all solutions to(57) align exponentially fast at a rate functionally dependent on Ψ, initial data, and ε,

maxα,i|vαi(t)−V| . e−δt.

22 ROMAN SHVYDKOY

Asymptotic dependence of the implied alignment rates for small ε and large λα for the Cucker-Smalekernel can be readily obtained from formulas (29) and (30). Thus, in the context of fast local alignment we

obtain δ ∼ λα for all β ≤ 1, while in the context of slow alignment, δ ∼ ε 11−β , for β < 1, and δ ∼ εe−1/ε for

β = 1.It is sometimes convenient to pass to the reference frame evolving with the momentum and center of mass

of the flock:

(61) wαi = vαi −Vα, yαi = xαi −Xα.

Using (57) and (58) one readily obtains the system

(62)

yαi = wαi,

wαi = λα

Nα∑j=1

mαjφαij(wαi −wαj)− εRα(t)wαi,

where

(63) Rα(t) =∑β 6=α

MβΨ(Xα −Xβ).

We used a shortcut to denote φαij = φα(yαi − yαj). The following lemma is straightforward.

Lemma 3.14. The old set of variables (xαi,vαi)α,i satisfies (57) if and only if the new set (yαi,wαi)α,isatisfies (62) and the macroscopic variables (Xα,Vα)α satisfy (58).

An immediate consequence of (62) is the maximum principle within each flock relative to its momentum.In particular we obtain a class of solutions, called Mikado flocks, given by

wαi(t) = wαi(t)rα,

where rα are fixed unit vectors. We will address those in more detail in the context of hydrodynamic systemsin Section 10.2.

Another type of cluster-adapted communication appears in multi-species models. Here, each flock αrepresents a cluster of species where communication between agents inside each cluster is homogeneous asin classical CS, and communication between species is regulated by a different set of kernels:

(64) vαi =

A∑β=1

1

Nβ

Nβ∑j=1

φαβ(xαi − xβj)(vβj − vαi).

One can easily run the same scheme and prove alignment with a rather rough condition: the heavy tail ofthe kernel φ(r) = minα,β φαβ(r). A more subtle spectral condition can be obtained in terms of the weightedgraph Laplacian associated with the matrix kernel Φ = (φαβ):

(65) (∆Φ(r))αβ =

−φαβ(r)

√MαMβ , α 6= β,∑

γ 6=αMγφαγ(r), α = β.

Namely, the condition requires the second eigenvalue of ∆Φ(r), denoted λ2(∆Φ(r)), to have heavy tail:

λ2(∆Φ(r)) &1

(1 + r)1−δ .

Then the energy of fluctuations of each flock converges to zero exponentially fast: V2(α) . e−δt. We referthe reader to [50] for details.

3.9. Notes and References. System (4) with smooth kernel (3) was first introduced by Cucker and Smalein [33, 34] where Theorem 3.3 was established using graph theoretical approach. This was one of the firstmodels which featured unconditional alignment, which is partly responsible for its success. Detailed surveysare given in [1, 97, 72, 69]. Shortly after, an analytical proof of the more general Theorem 3.4 was found byHa and Liu, [46], which has undergone several iterations in literature. Stability estimates appeared in Ha,Kim and Zhang [45]. We will return to them recursively in kinetic and hydrodynamic contexts.


Collision avoidance in the context of CS systems was first addressed in Park at al [76] with the use of aninter-particle bonding force showing no collision for 2 agent configurations. Cucker and Dong [31] proposeda simpler model with unconditional avoidance incorporating a singular repulsion force, see Chapter 4 below.For unforced systems the use of singular communication was proposed by Carrillo, Choi, Mucha and Peszek[17]. The result has been extended to models with nonlinear couplings by Markou, [68], and to variousforced systems by Peszek et al [58, 26]. For weakly singular kernels β < 1 despite presence of collisions,Peszek developed a theory of weak solutions to the system (4), [77, 78]. The estimates (39)–(41) are slightimprovements over [17] suitable for applications to the corrector method, which was introduced in Dietertand Shvydkoy [38] as a way to handle degenerate coercivity. For local kernels flocking dynamics can alsobe achieved by steering the system with a centralized control, see Capnigro et al [12]. Moreover, any givencollective outcome can also be achieved using decentralized control algorithm, [26].

Motsch-Tadmor model was first introduced in [71] and subsequently appeared in several applications, see[72]. It plays a further role in passing to hydrodynamic limit in multi-scale models to recover pressurelessEuler alignment system, which we discuss in detail in later chapters. Multi-flocks are introduced in [88] withMikado solutions presented in [63]. Multi-species models appeared earlier in [50].

24 ROMAN SHVYDKOY

4. Forced systems

The Cucker-Smale alignment force is typically supplemented by other forces in more realistic complexmodels. The theoretical studies of animal behavior distinguish three basic regions of mutual interac-tions in self-organized system. The first inner region is dominated by repulsion forces, driven by theneed to avoid collisions. Second intermediate region is characterized by alignment relative to the direc-tion of motion of other objects, which is where Cucker-Smale force plays the dominant role. And the thirdouter region is characterized by tendencies to attract and stay within the group. This simplified 3Zonemodel of interactions illustrated in Figure 5 has been fundamental in many studies in behavioral biology[3, 30, 52, 20, 22, 21]. A notable implementation of the 3Zone model to computer animation of birdsflight was proposed by Reynolds in [82], which was subsequently applied in motion picture production.

repulsion

alignment

attraction

Figure 5. 3Zone model

More refined modeling such as inclusion of cone of vision intocommunication zone, nearest neighbor rule, self-propulsion,friction, noise, external confinement allows to replicate some ofthe interesting behavioral phenomena such as milling patterns,aggregation, etc. A model with a wealth of such outcomes wasproposed by D’Orsogna et al [27]:

xi = vi,

vi = (α− β|vi|2)vi −1

N

N∑j=1

∇U(xi − xj),

We refer to [13, 15] for comprehensive surveys on models withattraction/repulsion forces.

The purpose of this chapter is to study the effects of theseadditional forces in the context of alignment models and seehow flocking dynamics changes or persists. The center ofour attention will be attraction/repulsion potential forces andthe method of hypocoercivity which allows to produce N -independent estimates suitable for large crowd limit N → ∞.

The most well understood case here is the 2Zone attraction/alignment model. In presence of repulsion or full3Zone model only partial N -dependent results are known. Much of the difficulty of the 3Zone model boilsdown to non-existence of naturally aligned steady states that minimize total energy. Say, if U ≥ 0 is a radialpotential with U = 0 in the alignment annulus |x| ∈ [r0, r1], then all separations of such stationary solutionswill need to fit within it, |xij | ∈ [r0, r1]. This is geometrically impossible unless N ≤ N(r0, r1, n). For thisreason we do not expect to have any strong flocking behavior. The repulsion itself acts as a mechanism ofexpansion of the flock which also presents an obstacle in obtaining N -dependent bounds on the diameter. So,we will proceed being mindful of these mathematical limitations and start with establishing general flockingresults for 3Zone models.

4.1. 3Zones model. Small crowd flocking. Let us consider Cucker-Smale system with a general potentialinteraction force

(66)

xi = vi,

vi =1

N

N∑j=1

φ(xi − xj)(vj − vi)−1

N

N∑j=1

∇U(xi − xj),

Here we can work with a rather general radially symmetric potential U ∈ C1 as long as it forms singularityat the origin and infinity

(67) limr→0,∞

U(r) =∞.

We also assume that

(68) φ ∈ C1, φ(r) > 0 for all r > r0, limr→∞

φ(r) = 0.

Although necessarily there is an overlap of alignment and attraction ranges we do not assume any specificrate. So the kernel can decay as fast as possible as long as it remains positive in the long range. In other


repulsion alignment attraction

Figure 6. Alignment versus potential in 3Zone model

words, alignment action is mostly confined to an intermediate region which is consistent with its placementin the 3Zone model. Figure 6 illustrates a typical picture we have in mind.

The system (66) is Galilean invariant and conserves momentum. So, as always we can assume v = 0. Thetotal energy is now composed of kinetic and potential parts

E = K + P,

K =1

2N

N∑i=1

|vi|2 =1

4N2

N∑i=1

|vij |2, P =1

N2

N∑i,j=1

U(xij),(69)

We have the energy law

(70)d

dtE = −I2.

In particular it implies P remains bounded which in view of (67) implies that that flock remains bounded,D(t) ≤ DN , and non-collisional. Note that the upper bound depends on N , so such conclusions wouldnot hold in the large crowd limit N → ∞. That is why we consider them more suitable for small crowdapplications.

The energy law above is lacking coercivity for two reasons – it is missing the potential energy entirelyon the right hand side, and the communication bound φ(DN ) may in fact vanish due to degeneracy of thekernel. In the next two sections we will demonstrate how the potential energy can be recovered in 2Zonecase when only attraction is present. In general we can only fill in the missing communication using ourcorrector method developed in Section 3.7.

Indeed, let us recall the estimate (50). In its derivation we have not used the equation except the maximumprinciple. In view of the global bound on K we still have a bound on each velocity |vi| < CN by a constantdepending on N . So, the only difference for us will be in constant a which now depends on N . We thereforearrive at ∫ ∞

0

Φ(D)V4(t) dt <∞.Given that Φ > 0 and D is bounded we conclude that∫ ∞

0

V4(t) dt <∞.

It is straightforward to see also that |ddtV4| ≤ CN . So, V4 is in fact a uniformly continuous function. Together

with integrability it implies that V4 → 0, and hence the system aligns.As noted in the introduction, it is unlikely that substantially faster convergence can be proved in general,

as the system is not likely to flock strongly.

Theorem 4.1. Any 3Zone model (66) with potential satisfying (67) and kernel satisfying (68) aligns A(t)→0, and flocks D(t) ≤ DN , although with rates and bounds which may depend on N .

Even if the kernel φ was non-degenerate we could have not improved the result. The natural way wouldbe to operate with the energy law

d

dtE ≤ −c0K

26 ROMAN SHVYDKOY

and conclude that∫∞

0K(t) dt < ∞. This is still insufficient for strong flocking since the residual velocities

would stir the system ever so slightly. The force can be modified to avoid this issue by having an additionalkinetic factor

(71) Fi = −√V2

1

N

N∑j=1

∇U(xi − xj).

This model was considered by Cucker and Dong [31] as a way to obtain strong flocking and collision avoidancein repulsion-only setting, SuppU ⊂ r < r0. One can argue that it is in fact natural for repulsive forcesto subside for still crowd since there is no reason for agents to panic if they are not on a collision course.The same perhaps cannot be said for the attraction force since the tendency to stay together would inducemotion even in a still but dispersed flock. Nonetheless, let us discuss this kinetic forcing in general.

Here we assume that communication is non-degenerate and either U →∞ as r →∞ or φ has a monotoneheavy tail. The energy law reads

d

dt(√K + P) = − I√

K.

So, in the first case we conclude boundedness of the diameter as earlier. In the second, we obtain∫ ∞0

φ(D(t))√K(t) dt <∞.

Noting that ddtD ≤ CN

√K(t), we further obtain∫ D(t)

D0

φ(r) dr <∞.

This immediately implies D(t) ≤ DN . With bounded diameter we draw a conclusion that the kinetic energyis in L1/2(R+): ∫ ∞

0

√K(t) dt <∞.

In particular, this implies that A is integrable and as a consequence, strong flocking ensues. ConvergenceA → 0 follows as before from uniform continuity.

Theorem 4.2. Consider the system (66) with non-degenerate φ > 0 and either strong potential U(r) →∞, r → ∞, or monotonely decaying heavy tail φ. Then any solution aligns A → 0, and flocks strongly,xi(t)− xj(t)→ xij.

Let us note that the geometric obstruction to flocking we alluded to earlier does not apply in this casesince the system has arbitrarily dispersed fully aligned steady states.

4.2. External confinement. Hypocoercivity. In this and next sections we will consider 2Zone modelswhere non-degenerate alignment is combined with potential attraction forces. The goal will be to showthat such models exhibit much more robust flocking properties with the results being N -independent andextendable to macroscopic models as well.

Let us first illustrate the method on a simpler example of a confinement force

(72)

xi = vi,

vi =1

N

N∑j=1

φ(xi − xj)(vj − vi)−∇U(xi),

where U is a convex radially increasing potential. While in general one can perform analysis in the case ofa strictly convex potential, for the same of brevity we focus on one particular case of quadratic U :

(73) U(x) =1

2|x|2.


In this case, the system reads

(74)

xi = vi,

vi =1

N

N∑j=1

φ(xi − xj)(vj − vi)− xi,

The natural limiting state of this system is not, in fact, alignment, but rather a harmonic oscillator.Indeed, denoting as before x, v the center of mass and momentum, we obtain

(75)d

dtx = v,

d

dtv = −x.

Also due to the linear nature of the forcing we can shift the system of coordinates to (x, v) and assume thatx = 0, v = 0. In this case system (74) is technically equivalent to the system with interaction forces (66)with the same quadratic potential. So, the question becomes to show that the solution tends to zero in thenew reference frame (still denoted (x,v)).

The full energy of the system is now given by

E = K + P

K =1

2N

N∑i=1

|vi|2, P =1

2N

N∑i=1

|xi|2.(76)

It will also be important to consider the “particle energy”, i.e. L∞-version of the energy:

(77) E∞ =1

2maxi

(|vi|2 + |xi|2).

Theorem 4.3. Suppose that the kernel φ is bounded, decreasing, and satisfies the weak heavy tail condition

(78)

∫ ∞0

rφ(r) dr =∞.

Then the system (74) settles on its harmonic oscillator (75) exponentially fast, meaning

(79) maxi

(|vi(t)− v(t)|2 + |xi(t)− x(t)|2) ≤ Ce−δt,

for some δ > 0 and C = C(v0,x0, φ) independent of N .

Proof. The result amounts to establishing an exponential bound on E∞.Straight from the equations we obtain the energy law

(80)d

dtE = −I2 ≤ −φ(D)K.

Note that the dissipation is not coercive even if we knew a lower bound φ(D) > c0. However, we proceedwith E∞:

d

dtE∞ ≤

1

N

∑j

φij(vj − vi) · vi ≤1

2N

∑j

φij(|vj |2 − |vi|2) ≤ 1

2N

∑j

φij |vj |2 ≤ |φ|∞K.

To combine the two equations into one system, let us note that D ≤ 4√E∞. Thus,

d

dtE ≤ −φ(4

√E∞)K

d

dtE∞ ≤ CK.

(81)

Consider the Lyapunov function

L = E +1

C

∫ E∞0

φ(4√r) dr.

From our heavy tail assumption it follows that E∞ remains bounded, from which we conclude that D(t) ≤ D.Now the energy law (80) reads

(82)d

dtE ≤ −c0K.

28 ROMAN SHVYDKOY

Next we proceed with a hypocoercivity argument to restore K on the right hand side to the full energy E .We consider the corrector (longitudinal momentum)

(83) X =1

N

N∑i=1

xi · vi,

and note that for λ < 14 , E + λX ∼ E . Let us compute the derivative:

d

dtX =

1

N2

∑j,i

φij(vj − vi) · xi +1

N

∑i

(|vi|2 − |xi|2)

≤ |φ|∞N2

∑j,i

(4|vj |2 + 4|vi|2 +1

2|xi|2) +

1

N

∑i

(|vi|2 − |xi|2) ≤ CK − 1

4P.

Choosing λ < c0/2C we obtain

d

dt(E + λX ) ≤ −c1E ∼ −(E + λX ).

This establishes exponential decay of E .Now that the L2-energy is decaying exponentially, we can extrapolate to obtain exponential decay for E∞

as well. The method is similar – we consider an amended version of E∞:

(84) Eλ∞ =1

2maxi

(|vi|2 + |xi|2 + λxi · vi).

Again, for λ < 14 , this does not alter the particle energy much, E∞ ∼ Eλ∞. Differentiating at a point of

maximum we obtain

d

dtEλ∞ ≤

1

N

∑j

φij(vj − vi) · (vi + λxi) + λ|vi|2 − λ|xi|2

≤ 1

N

∑j

φijvj · vi + λ1

N

∑j

φijvj · xi −1

N

∑j

φij |vi|2 − λ1

N

∑j

φijvi · xi + λ|vi|2 − λ|xi|2.

In the gain term − 1N

∑j φij |vi|2 we replace φij by a lower bound c0:

− 1

N

∑j

φij |vi|2 ≤ −c0|vi|2.

and in the rest we simply use boundedness of the kernel. Hence, if λ is small enough , the term λ|vi|2 getsabsorbed. We obtain at this point

d

dtEλ∞ .

1

N

∑j

|vj ||vi|+ λ1

N

∑j

|vj ||xi|+ λ1

N

∑j

|vi||xi| − c1|vi|2 − λ|xi|2

≤ 4K +c14|vi|2 + 4λK +

1

4λ|xi|2 + λ|vi|2 +

1

4λ|xi|2 − c1|vi|2 − λ|xi|2

. K − |vi|2 − |xi|2 . K − Eλ∞.

Since we already know that K is exponentially decaying, this establishes a similar bound on Eλ∞.

Let us note that for small number of agents, when the dependence on N is not an issue, one can actuallyobtain a much stronger result: as long as φ(r) > 0 for all r > 0, the conclusion of the theorem holds true.Indeed, from the first lines we have established that the total energy is decaying and, hence, bounded. Butfrom the potential part P this immediately implies that the flock is bounded, with a bound depending onN . This sets the rest of the argument to go through.

This observation clearly shows that it is impossible to construct a simple example with a few agents, likewe did in Example 3.1, to prove sharpness of the heavy tail condition (78).


φ(r) U(r)

Alignment

b

LAttraction

Figure 7. 2Zone Attraction-Alignment model

4.3. 2Zone model: attraction + alignment. We consider the system with pairwise interactions deter-mined by a radially symmetric smooth potential U ∈ C2(R+):

(85)

xi = vi,

vi =1

N

N∑j=1

φ(xi − xj)(vj − vi)−1

N

N∑j=1

∇U(xi − xj).

Notice first that the system (85) preserves momentum and is Galilean invariant. So, we can shift thecenter of mass and momentum to zero. Let us denote the forces by

Fi =1

N

N∑j=1

∇U(xi − xj), F = (F1, . . . ,FN ), |F|22 =1

N

N∑j=1

|Fi|2.

The energy still satisfies the classical law:

(86)d

dtE = − 1

2N2

N∑i,j=1

φij |vij |2 = −I.

In this section we consider an Attraction-Alignment 2Zone model with non-degenerate communicationkernel:

(87) φ′(r) ≤ 0, φ(r) ≥ c0〈r〉γ , for r ≥ 0.

For the potential we assume essentially a power law: for some β > 1 and L′ > L > 0,

Support: U ∈ C2(R+), U(r) = 0, ∀r ≤ L,Growth: U(r) ≥ a0r

β , |U ′(r)| ≤ a1rβ−1, |U ′′(r)| ≤ a2r

β−2, ∀r > L′,

Convexity: U ′(r), U ′′(r) ≥ 0, ∀r > 0.

(88)

Theorem 4.4. Under the assumptions (87) and (88) on the kernel and potential in the range of parametersgiven by

(89) γ <

1, 1 < β <

4

3,

3

2β − 1,

4

3≤ β < 2,

2, β ≥ 2,

all solutions to the system (85) flock

D(t) ≤ D <∞,and align

(90) E(t) ≤ Cδ〈t〉1−δ , ∀δ > 0.

30 ROMAN SHVYDKOY

Proof. We will operate with the particle energy defined similarly to the confinement case:

(91) Ei =1

2|vi|2 +

1

N

N∑k=1

U(xik), E∞ = maxiEi.

First, let us observe that the particle energy controls the diameter of the flock. By convexity and ourassumptions on the growth of the potential, we have

(92) Ei ≥ U(xi) ≥ (|xi| − L′)β+.So,

(93) D ≤ E1/β∞ + L′.

Let us now establish a bound on E∞. For each i we compute

(94)d

dtEi =

1

N

N∑k=1

φikvki · vi −1

N

N∑k=1

∇U(xik) · vk.

For the kinetic part we use the identity

(95) vki · vi = −1

2|vki|2 −

1

2|vi|2 +

1

2|vk|2.

Discarding all the negative terms, we bound

1

N

N∑k=1

φikvki · vi ≤ |φ|∞K.

Due to the energy law, K, of course will remain bounded, but we will keep it for now. As to the potentialterm, there are several ways we can handle it.

For any 1 ≤ β ≤ 43 we can derive a direct estimate from the first derivative:∣∣∣∣∣ 1

N

N∑k=1

∇U(xik) · vk∣∣∣∣∣ ≤ √K

(1

N

N∑k=1

|∇U(xik)|2) 1

2

≤√KDβ−1.

Consequently,d

dtE(i) ≤ c1K + c2

√KDβ−1 .

√K(1 + E

β−1β∞ ),

and

(96)d

dtE∞ ≤ c3

√K(1 + E

β−1β∞ ) ⇒ E∞ . 〈t〉β ⇒ D . 〈t〉.

In the range 43 ≤ β ≤ 2 it is better to make use of the second derivative:∣∣∣∣∣ 1

N

N∑k=1

∇U(xik) · vk∣∣∣∣∣ =

1

N

N∑k=1

(∇U(xik)−∇U(xi)) · vk ≤ ‖D2U‖∞√K(

1

N

N∑k=1

|xk|2) 1

2

≤ c4√K

1

N2

N∑i,j=1

|xij |2 1

2

.

(97)

The following inequality will be used repeatedly

(98)1

N2

N∑i,j=1

|xij |2 ≤ (L′)2 +1

N2

N∑i,j=1

(|xij | − L′)2+ ≤ C(1 +D(2−β)+P).

Continuing the above,∣∣∣∣∣ 1

N

N∑k=1

∇U(xik) · vk∣∣∣∣∣ ≤ c4√K(1 +D2−βP)1/2 ≤ c5

√K(1 + E∞)

2−β2β .


In this case,

(99)d

dtE∞ ≤ c6

√K(1 + E∞)

2−β2β ⇒ E∞ . 〈t〉

2β3β−2 ⇒ D ≤ 〈t〉 2

3β−2 .

Finally, for β > 2, we argue similarly, using that |D2U(xik)| ≤ Dβ−2, and (98), to obtain∣∣∣∣∣ 1

N

N∑k=1

∇U(xik) · vk∣∣∣∣∣ ≤ √KDβ−2,

and hence,

(100)d

dtE∞ ≤ c7

√K(1 + E∞)

β−2β ⇒ E∞ . 〈t〉

β2 ⇒ D ≤ 〈t〉 1

2 .

We have proved the following a priori estimate:

(101) D(t) . 〈t〉d, where d =

1, 1 ≤ β < 4

3,

2

3β − 2,

4

3≤ β < 2,

1

2, β ≥ 2.

Denoteζ(t) = 〈t〉−γd.

According to the basic energy equation (86) we have

(102)d

dtE ≤ −1

2I − cζ(t)K.

Considering this as a starting point, just like in the quadratic confinement case, we will build correctors tothe energy to achieve full coercivity on the right hand side of (102). We introduce one more auxiliary powerfunction

η(t) = 〈t〉−α, γd ≤ α < 1.

First, we consider the same longitudinal momentum

X =1

N

N∑i=1

xi · vi.

It will come with a prefactor εη(t), where ε is a small parameter. Let us estimate using (98):

εη(t)|X | ≤ εK + εη2(t)1

N2

N∑i,j=1

|xij |2 ≤ εK + cεη2(t) + εη2(t)D(2−β)+P.

The potential term is bounded by εP as long as

2α ≥ d(2− β)+.

Hence,

(103) εη(t)|X | ≤ εE + cη2(t).

This shows thatE + εη(t)X + 2cη2(t) ∼ E + cεη2(t).

Let us now consider the derivative

X ′ =1

N2

N∑i=1

|vi|2 +1

N2

N∑i,k=1

xik · vkiφki −1

N2

N∑i,k=1

xik · ∇U(xik) = K +A−B.

The gain term B, by convexity dominates the potential energy B ≥ P. This is the main reason why weintroduced the X -corrector. As to A:

|A| ≤ |φ|∞2ε1/2η(t)

I +ε1/2η(t)

2

1

N2

N∑i,j=1

|xij |2 .1

ε1/2η(t)I + ε1/2η(t) + ε1/2η(t)D(2−β)+P.

32 ROMAN SHVYDKOY

By requiring a more stringent assumption on parameters

(104) α ≥ d(2− β)+,

we can ensure that the potential term is bounded by ∼ ε1/2P, which can be absorbed by the gain term. Sofar, we have obtained

(105)d

dt(E + εη(t)X + 2cη2(t)) ≤ −c1εη(t)E + c2η

2(t) + εη′(t)X .

In view of (103),

|εη′(t)X| ≤ ε 1

〈t〉η(t)|X | ≤ ε 1

〈t〉E + εη2(t)

〈t〉 .

Since α < 1, the energy term will be absorbed, and the free term is even smaller then η2. Denoting

E = E + εη(t)X + 2cη2(t),

we obtaind

dtE ≤ −c1η(t)E + c2η

2(t).

By Duhamel’s formula,

E(t) . exp−〈t〉1−α+ exp−〈t〉1−α∫ t

0

e〈s〉1−α

〈s〉2α ds.

By an elementary asymptotic analysis,∫ t

0

e〈s〉1−α′

〈s〉α′′ ds ∼ exp〈t〉1−α′ 1

〈t〉α′′−α′ .

Thus, we obtain an algebraic decay rate

(106) E(t) .1

〈t〉α , ∀α < 1,

provided

(107) dγ < 1 and d(2− β)+ < 1.

This translates exactly into the conditions on γ given by (89), and (106) automatically implies (90)Going back to the estimates (96) and (99), but keeping the kinetic energy with its established decay, we

obtain a new decay rate for the diameter

D ≤ Cδ〈t〉d2 +δ, ∀δ > 0.

At the next stage we prove flocking: D(t) < D. In order to achieve this we return again to the particleenergy estimates. Let us denote

Pi =1

N

N∑k=1

U(xik), Ii =1

N

N∑k=1

φik|vki|2, Xi = xi · vi.

Using (94), (95), (97), (98) and the fact that D(2−β)+P has a negative rate of decrease, we obtain

d

dtEi ≤ K −

1

2φ(D)|vi|2 − Ii + c

√K . −1

2φ(D)|vi|2 − Ii +

1

〈t〉 12−δ

, ∀δ > 0.

In view of (107), we can pick α and δ such that

dγ

2+ δγ <

1

2− 2δ < α <

1

2− δ

(2− β)+d+ 2δ(2− β)+ < 2α.(108)

We use as before the auxiliary rate function η(t) = 〈t〉−α. Let us estimate the corrector

|εη(t)Xi| ≤ ε|vi|2 + εη2(t)|xi|2 ≤ ε|vi|2 + εη2(t)D2−βPi + L2εη2(t) ≤ ε|vi|2 + cεPi + L2εη2(t).

So,

Ei := Ei + εη(t)Xi + 2L2εη2(t) ∼ Ei + L2εη2(t).


Differentiating,

X ′i = |vi|2 +1

N

N∑k=1

xi · vkiφki −1

N

N∑k=1

xik · ∇U(xik) +1

N

N∑k=1

xk · (∇U(xik)−∇U(xi))

≤ |vi|2 + ε1/2η(t)|xi|2 +1

ε1/2η(t)Ii − Pi +

1

N2

N∑l,k=1

|xkl|2

≤ |vi|2 + ε1/2L2η(t) + ε1/2D(2−β)+η(t)Pi +1

ε1/2η(t)Ii − Pi + C

in view of (108), ε1/2D(2−β)+η(t) . ε1/2, so the potential term is absorbed by −Pi,

≤ |vi|2 +1

η(t)Ii −

1

2Pi + C.

Again in view of (108), η(t) decays faster than φ(D), so plugging into the energy equation we obtain

d

dtEi ≤ −εη(t)Ei + η(t) +

√K + εη′(t)Xi,

and as before εη′(t)X is a lower order term which is absorbed in the negative energy term and +η2. So,

d

dtEi ≤ −εη(t)Ei + η(t) +

√K.

By our choice of constants (108),√K decays faster than η(t), hence,

d

dtEi . −εη(t)Ei + η(t).

This proves boundedness of Ei, and hence that of Ei + L2εη2(t), and hence that of Ei. In view of (93), thisimplies flocking:

(109) D(t) < D, ∀t > 0.

It is interesting to note that when the support of the potential spans the entire line, L = 0, and U landsat the origin with at least a quadratic touch:

(110) U(r) ≥ a0r2, r < L′,

then we can establish exponential alignment in terms of the energy E . Indeed, since we already know thatthe diameter is bounded, the basic energy equation reads

d

dtE ≤ −c0K −

1

2I.

The momentum corrector needs only an ε-prefactor to satisfy the bound

|εX| ≤ εK + εcP.This is due to the assumed quadratic order of the potential near the origin and, again, boundedness of thediameter. Hence, E + εX ∼ E . The rest of the argument is similar to the confinement case. We obtain

X . K + ε1/2P +1

ε1/2I − P ≤ K − 1

2P 1

ε1/2I

Thus,d

dt(E + εX ) ≤ −c1E ∼ −c1(E + εX ).

This proves exponential decay of E . Going further to consider the individual particle energies, we discoversimilar decays. Indeed, denoting by Exp any quantity that decays exponentially fast, we follow the samescheme:

d

dtEi ≤ −c1|vi|2 −

1

2Ii + Exp.

In view of |xi|2 . Pi,ε|Xi| ≤ ε|vi|2 + εPi,

34 ROMAN SHVYDKOY

so Ei + εXi ∼ Ei. Further following the estimates as in the proof,

X ′i . |vi|2 +1

ε1/2Ii −

1

2Pi.

Thus,d

dt(Ei + εXi) ≤ −c1(Ei + εXi) + Exp.

This establishes exponential decay for E∞, and hence for the individual velocities. This also proves thatD(t) = Exp. So, the alignment outcome here is exponential shrinking a point.

Theorem 4.5. Let us assume that the support of the potential spans the entire space and (110). Then thesolutions flock and align exponentially fast:

D(t) + |v(t)− v|∞ ≤ Ce−δt,for some C, δ > 0.

4.4. Dynamics under self-propulsion and Rayleigh friction. In this section we investigate how dy-namics of the Cucker-Smale system changes under an addition of a balanced self-propulsion and Rayleighfriction forces given by

Fi = σvi(1− |vi|2),

where σ > 0 is a strength parameter. So, the system reads

(111)

xi = vi,

vi = λ

N∑j=1

mjφ(xi,xj)(vj − vi) + σvi(1− |vi|2).

Clearly it drives all magnitudes |vi| to the same value 1. While it might seem to be another alignmentmechanism, it may in fact counteract the Cucker-Smale force provided the velocities are pointing in differentdirection. Another point of caution while working with friction forces is that the system no longer conservesthe momentum or is Galilean invariant. Let us first establish a few unconditional facts about the long timebehavior of the new system.

First let us compute evolution of m = maxi |vi|. Via the usual maximizing functional approach we obtain

d

dtm ≤ σm(1−m2).

Solving this ODI directly we obtain

m(t) ≤ et√c20 + e2t

,

where c0 depends on the initial condition only. So, it is clear that the friction acts strongly on largemagnitudes to bring all vectors exponentially fast to the unit ball,

(112) |vi(t)| ≤ 1 +O(e−t).

Thus, the product ΠNi=1B1(0) is an absorbing ball of the system.

Unfortunately it is not always that case that all vectors converge to absolute value 1 as a result ofcounteraction of the alignment. Let us consider the following illuminating examples. Let us assume that wehave a global communication φ ≡ 1, and consider a two agent system on the line where v = v1 = −v2 > 0.Then we have the system

x = v, v = −λv + σv(1− v2).

The equation can be solved explicitly. If the Cucker-Smale communication is weak, λ < σ, then the solutionis given by

v =

√1− λ

σ√1 + c20e

−2t(σ−λ).

So, as we can see even global communication is not sufficient to provide alignment in this case.When λ = σ, we obtain

v(t) =v0√

2σtv20 + 1

.


Hence, the solution aligns but weakly, and it does not converge to the natural value v = 1. At the same timewe can see that the agents diverge, x(t) ∼

√t. So, no flocking occurs either.

Finally, when λ > σ we obtain a positive alignment result

v =c0e

(σ−λ)t√

λσ − 1√

1− c20e2t(σ−λ).

So, in this case v → 0 exponentially fast and flocking ensues.The latter case represents a general result which we state next.

Lemma 4.6. Let φ∗ = inf φ > 0 and λφ∗M > σ. Then the solution aligns and flocks exponentially fast:

A ≤ A0e(σ−λφ∗M)t.

Proof. We argue as in the proof of Theorem 3.4 where we replace the entire kernel with its minimal value:

d

dtA ≤ −λφ∗MA+ σ`[vi(1− |vi|2)− vj(1− |vj |2)].

Here (`, i, j) is a maximizing triple for A. Considering the functional

G(w) = w(1− |w|2),

we findDwG(w) = Id−|w|2 Id−2w ⊗w.

Thus,vi(1− |vi|2)− vj(1− |vj |2) = DwG(w)(vi − vj),

for some w on the segment [vi,vj ]. Considering that ` =vi−vj|vi−vj | we can dismiss the entire negative definite

part of DwG, with the remaining part being the identity. So, we obtain

d

dtA ≤ −λφ∗MA+ σA.

This finishes the proof.

Under the conditions of the last theorem we can in fact deduce much more precise information about thelong time dynamics. Let us denote by E = E(t) any exponentially decaying function. We have so far

vi = σvi(1− |vi|2) + E.

Multiplying by vi and denoting y = |vi|2 we obtain the following ODE

(113) y = 2σy(1− y) + E.

Although the pure logistic equation y = 2σy(1− y) is easy to solve – all positive solutions converge to 1 orstay 0 if initially so – the analysis of the forced ODE requires elaboration. Let us keep in mind that we havea solution y that is a priori non-negative. We also know from (112) that it gets attracted into [0, 1].

Lemma 4.7. Any non-negative solution to (113) either converges to 0 or to 1. In the latter case, convergenceoccurs exponentially fast.

Proof. Indeed, suppose y does not converge to 0. Then there exists a δ > 0 for which there exists a sequenceof times t1, t2, ...→∞ such that y(ti) > δ. For ti large enough we have

2σδ(1− δ) + E(ti) > 0.

Therefore, from that time on, y(t) will never cross δ again, y(t) > δ, t > ti. We have then

d

dt(1− y) = −2σy(1− y) + E,

and in view of ddt

(1− y)+ = 11−y>0ddt

(1− y) in distributional sense, we obtain

d

dt(1− y)+ = −2σy(1− y)+ + E ≤ −2σδ(1− y)+ + E.

By Gronwall’s lemma, (1− y)+ → 0 exponentially fast, which in view of (112) implies y → 1 exponentiallyfast.

36 ROMAN SHVYDKOY

Since A → 0, we conclude from the lemma that either all vi → 0 or all |vi| → 1 exponentially fast. Inthe latter case we conclude that vi = E, and hence all vi’s must converge to a single vector on Sn−1. Wetherefore have a full description of the dynamics under strong alignment.

Theorem 4.8. Let φ∗ = inf φ > 0 and λφ∗M > σ. Then A → 0 exponentially fast, and either all vi → 0or there exists a vector v ∈ Sn−1 to which all vi converge exponentially fast.

We can see that since 0 is an unstable solution to the pure logistic equation, convergence to zero issomewhat unlikely scenario for solutions of the system. In the example presented above this however wasthe case in view of the special symmetric configuration v1 = −v2, which of course ensures exponentialconvergence due to Lemma 4.6. It turns out that 0 as a limit can be eliminated for a class of examples withproperties opposite to the one we considered above, namely, for solutions that belong to coordinate sectors:

Σθ1,...,θn = (v1, . . . ,vN ) : sgn(vki ) = θk,∀i = 1, . . . , N.First we observe that these sectors remain invariant. Indeed, it is clear that all the coordinate planesπk = vki = 0,∀i = 1, . . . , N remain invariant. Next, the system is invariant under one sign inversion:

vki → −vki , xki → −xki ,

for a fixed k. So, we can focus on one case only Σ = Σ1,...,1. If we start initially in Σ, let us denote by t∗

the first time any coordinate turns into 0. Let vki (t∗) = 0. Since we are not on the plane πk, not all of theother agent’s kth coordinate vanishes: ∃vkj (t∗) > 0. Then,

vki (t∗) =∑j

mjφijvkj > 0,

which is a contradiction.

Theorem 4.9. Let φ∗ = inf φ > 0 and λφ∗M > σ. For any initial condition in coordinate sector Σθ1,...,θn ,the solution converges exponentially fast to a vector v ∈ Sn−1.

Proof. Indeed, according to Theorem 4.8 we just need to eliminate the other alternative. So, suppose thatall vi → 0. We can also assume by invariance that the solution belongs to Σ. For each k let us findvki = minj vkj . Then

vki =∑j

mjφij(vkj − vki ) + σvki (1− |vi|2) ≥ σvki (1− |vi|2).

Since all |vi|2 → 0, from some point on, we find (1− |vi|2) > 12 . Then

vki ≥1

2σvki ,

which implies exponential growth, a contradiction.

4.5. Notes and References. Our Theorem 4.1 is new, although Kim and Peszek arrived at the same con-clusions for the quadratic attraction/repulsion potential U(r) = (r−R)2 and non-degenerate communicationφ earlier in [58]. Even though the potential is not singular at the origin one can still prove no collisionsstarting from some time T . Kinetic repulsion force was introduced by Cucker and Dong in [31] and in [32]generalized to non-linear couplings.

The N -independent analysis of 2Zone attraction models and hypocoercivity method was introduced byShu and Tadmor in recent works [86, 85]. The latter in fact addresses a more general class of Hamiltoniansystems with anticipation which lead to systems of Cucker-Smale type but with matrix communication. Weessentially follow their work with a minor modifications which allow to include a wider range of indexes β, γ,potentials extended to act from scale L, and Theorem 4.5 going one step further by establishing exponentialflocking in the case L = 0.

The problem which still remains unexplored in the context of repulsion/alignment or 3Zone models iswhether one can establish some flocking behavior for large crowds N →∞. This either amounts to findingN -independent estimates or working directly with the kinetic or hydrodynamics systems, see next chapters.

The Rayleigh friction with alignment was studied by Ha et al in [42], which we recast here in `∞-basedframework with a few corrections and added extensions.


5. Kinetic models

In the large crowd systems, where N ∼ ∞, it is more efficient to resort to mesoscopic level of descriptionof the Cucker-Smale dynamics. The corresponding kinetic formulation of (4) can be derived formally viathe BBGKY hierarchy. Looking slightly ahead we seek to derive the following Vlasov-type model whichdescribes evolution of a mass probability distribution f(x, v, t) of agents in phase space (x, v):

(114) ∂tf + v · ∇xf + λ∇v · [fF (f)] = 0,

where

F (f)(x, v, t) =

∫R2n

φ(x, y)(w − v)f(y, w, t) dw dy.

We will then proceed with rigorous derivation via the mean-field limit of empirical measures. Along the waywe establish contractivity and asymptotic stability estimates for heavy tail kernels.

5.1. BBGKY hierarchy: formal derivation. Let us consider a probability density

PN = PN (x1, v1, ..., xN , vN , t)

of a system of N agents in the ensamble configuration space

(x1, v1, ..., xN , vN ) ∈ R2nN .

The conservation of mass in the Gibbs ensemble propagated according to the given system (4) leads to theclassical Liouville equation:

(115) PNt +

N∑i=1

vi · ∇xiPN +

N∑i=1

∇vi · (viPN ) = 0.

We assume the effective radius of communication between agents remains independent of N , i.e. the kernelφ is not rescaled with N . This scaling regime called the mean-field limit. We further assume that thetotal mass M =

∑mi remains constant and maximi → 0. As a result, the agents become more and more

indistinguishable, which we reflect in the symmetry condition

PN (..., xi, vi, ..., xj , vj , ..., t) = PN (..., xj , vj , ..., xi, vi, ..., t).

We seek to derive an equation for the first marginal

P 1,N (x, v, t) =

∫R2n(N−1)

PN (x, v, x, v, t)dxdv,

where x = (x2, ..., xN ) and v = (v2, ..., vN ). Thus, integrating in x, v in (115) we obtain

P 1,Nt + v · ∇xP 1,N + λ∇v ·

∫R2n(N−1)

N∑j=2

mjφ(x, xj)(vj − v)PNdxdv = 0.

In view of the symmetry of PN , we achieve equality of the integrals in the sum above, and hence,

P 1,Nt + v · ∇xP 1,N + λ(M −m1)∇v ·

∫R2n

φ(x, y)(w − v)P 2,N (x, v, y, w, t) dy dw = 0,

where P 2,N is the second marginal:

P 2,N (x, v, y, w, t) =

∫R2n(N−2)

PN (x, v, y, w, ¯z, ¯u, t)d¯zd¯u.

Denoting the limiting densities by P = limN→∞ P 1,N , Q = limN→∞ P 2,N we obtain

Pt + v · ∇xP + λM∇v ·∫R2n

φ(x, y)(w − v)Q(x, v, y, w, t) dy dw = 0.

We close by making the molecular chaos assumption

Q(x, v, y, w, t) = P (x, v, t)P (y, w, t),

which results in precisely the following Vlasov-type equation (114) for the mass density f = MP .

38 ROMAN SHVYDKOY

5.2. Weak formulation and basic principles of kinetic dynamics. We now turn to rigorous studyand justification of the kinetic model (114). For simplicity this will be done under the assumption that thekernel φ is of convolution type, decreasing and smooth, φ ∈ C1.

A straightforward connection between discrete and kinetic models can be seen by considering the empiricalmeasure

(116) µNt =

N∑i=1

miδxi(t) ⊗ δvi(t),

which satisfies a weak formulation of (114) if and only if the system xi,vii solves the classical Cucker-Smale equations (4). This leads us first to define (114) in a weak sense for a special class of measure-valuedsolutions. We denote by M+(R2n) the set of non-negative Radon measures on R2n, and

Mp+ = µ ∈M+(R2n) : ‖µ‖p =

∫R2n

(1 + |v|p) dµ(x, v) <∞.

We endow M+ and well as Mp+ with the topology of weak convergence, which means convergence on

continuous bounded function Cb(R2n). Note that any bounded set in M2+ is automatically tight, i.e. for

every ε > 0 there exists a compact set Kε (a ball in fact) such that µ(R2n\Kε) < ε for all µ in the family.So, by Prohorov’s theorem weak convergence on Cb(R2n) is equivalent to the classical weak∗ convergence onC0(R2n), the predual of M(R2n). Therefore all bounded sets in M2

+ are weakly precompact. Since we willdeal with measures confined to a bounded set anyway, we will not detail the statements above here.

Let us note that for any µ ∈Mp+, p > 1 the integral

F (µ)(x, v) =

∫R2n

φ(x− y)(w − v) dµ(y, w)

defines a C1 smooth locally bounded field on R2n. Moreover, for a time-dependent weakly continuous familyµt0≤t<T ∈ Cw∗([0, T );Mp

+(R2n)), p > 1, the field F becomes continuous F (µ·) ∈ C([0, T ) × R2n) anduniformly Lipschitz F (µ·) ∈ L∞([0, T ); Liploc(R2n)) with

(117) |F (µt)(x, v)| ≤ C(1 + |v|).This is sufficient to define the proper global flow map on [0, T )× R2n later.

Definition 5.1. We say that µt0≤t<T ∈ Cw∗([0, T );M2+(R2n)) is a measure-valued solution to (114) with

initial condition µ0 if for any test-function g ∈ C∞0 ([0, T )× R2n) one has for all 0 < t < T

(118)

∫R2n

g(t, x, v) dµt(x, v) =

∫R2n

g(0, x, v) dµ0(x, v)+

+

∫ t

0

∫R2n

(∂sg + v · ∇xg + λF (µs) · ∇vg) dµs(x, v) ds.

The proof of the following lemma is straightforward.

Lemma 5.2. The empirical measure (116) satisfies the weak kinetic formulation (118) if and only if the setof pairs (xi,vi)i is a solution to (4).

The choice of p = 2 is motivated by our interest in the energy class, however any p > 1 is sufficient to makethe right hand side of (118) well defined. From compactly supported test functions, formulation (118) is easilyextendable to functions with quadratic growth of the material derivative: |∂sg|+ |v||∇xg|+ |v||∇vg| . |v|2.Immediate consequences are the mass conservation (plugging g = 1)

(119)d

dtµt(R2n) = 0,

conservation of momentum (plugging g = vi),

(120)d

dt

∫R2n

v dµt(x, v) = 0,

and the energy law (plugging g = |v|2),

(121)d

dt

∫R2n

1

2|v|2 dµt(x, v) = −λ

2

∫R2n

∫R2n

φ(x− y)|w − v|2 dµt(y, w) dµt(x, v).


One implication of the decaying energy is that any solution will automatically retain uniformly boundedsecond momentum on its interval of existence:

(122) ‖µt‖2 ≤ ‖µ0‖2.Another easy application is concentration towards macroscopic values in L2-sense, which presents a directanalogue of the discrete result (3.2). To make this precise, let us denote the average velocity

V =1

M

∫R2n

v dµt(x, v).

It determines the direction of the flock centered around the mass center

X =

∫R2n

xdµt(x, v),d

dtX = V .

Due to Galilean invariance of the equation (114)

µt → µt,

∫g(t, x, v)µt(x, v) =

∫g(t, x+X(t), v + V ) dµt(x, v)

we can always assume that V = X = 0. Assuming for the sake of simpler discussion that µt are given bydensity distributions f(x, v, t), we consider the macroscopic parameters

(123) ρ(x, t) =

∫Rnf(x, v, t) dv, ρu =

∫Rnvf(x, v, t) dv.

Then the total kinetic energy can be decomposed into internal (peculiar) and macroscopic energies as follows

E =1

2

∫R2n

|v|2f(x, v, t) dxdv = EI + Em

EI =1

2

∫R2n

|v − u(x, t)|2f(x, v, t) dx dv, Em =1

2

∫Rnρ(x, t)|u(x, t)|2 dx.

Note that due to the assumed zero momentum condition we have

Em =1

4M

∫R2n

ρ(x, t)ρ(y, t)|u(x, t)− u(y, t)|2 dx dy.

From (119) we obtaind

dtE ≤ −2φ(D(t))ME ,

where D = supx,y∈Supp f |x − y|. Arguing as in Section 3.4 we obtain a direct analogue of the discrete

statement in L2-terms:

(124) supt≥0D(t) ≤ D, E(t) ≤ E0e−2tλMφ(D).

Given the decomposition of the energy this expresses both tendency of the distribution f to the macroscopicvalues and global alignment of the macroscopic velocity u on the support of the flock. Remarkably, thelatter is derived without any knowledge of what macroscopic system u satisfies! We will discuss the closureproblem in more detail later.

To fully exploit the transport structure of the kinetic equation (114) let us consider the characteristic flowof the field 〈v, λF (µt)〉:

d

dtX(t, s, x, v) = V (t, s, x, v), X(s, s, x, v) = x(125)

d

dtV (t, s, x, v) = λF (µt)(X,V ), V (s, s, x, v) = v.(126)

We also denote X(t, 0, x, v) = X(t, x, v), V (t, 0, x, v) = V (t, x, v), and sometimes (x, v) = ω. Note that F (µt)is smooth in ω, so the flow is well-defined and smooth. In view of the linear bound (117) we also concludethat the ODE (125)–(126) is well-posed on the entire existence time interval of the solution µ. Using thetest-function g(s, ω) = h(X(t, s, ω), V (t, s, ω)) in (118), for some h ∈ C∞0 (R2n), we have

∂sg + v · ∇xg + λF (µs) · ∇vg = 0.

40 ROMAN SHVYDKOY

So, (118) reads

(127)

∫R2n

h(ω) dµt(ω) =

∫R2n

h(X(t, ω), V (t, ω)) dµ0(ω).

We say that µt is a push-forward of the measure µ0 under the flow-map (X,V ), µt = (X,V )#µ0. This willbe the key formula to deduce contractivity of the solution map to (114) and to study the mean-field limit.

Due to finite second momenta of the measures, (127) is extendable to |h| . |v|2 (note here a rough estimateon the velocity flow |V | ≤ C(t, s)(1 + |v|)). Hence, we can apply (127) to the V -equation (126) to rewrite itcompletely in Lagrangian coordinates

(128)d

dtV (t, ω) =

∫R2n

φ(X(t, ω)−X(t, ω′))(V (t, ω′)− V (t, ω)) dµ0(ω′).

In this form it presents a direct analogue to the discrete system (4), and as a result we can carry out all thebasic flocking results in a way very similar to the discrete case.

5.3. Kinetic maximum principle and flocking. It is clear from (128) that the velocity characteristicsare concentrating towards the mean value conserved in time. To quantify the rate of convergence stumblesupon the need for global communication in a way similar to the discrete case. Let us assume for now thatwe have a general non-negative kernel φ = φ(x− y) and work out a system of equation for the kinetic flockparameters. We define the amplitude and diameter over a general compact domain Ω containing the initialflock, Suppµ0 ⊂ Ω. This will serve multiple purposes – to show alignment on an arbitrarily wide domain,and to provide a tool to compare two different solutions. So, let us assume that we are working with a givensolution µ ∈ Cw∗([0, T );M2

+(R2n)) with finite initial support. We define

DΩ(t) = maxω′,ω′′∈Ω

|X(t, ω′)−X(t, ω′′)|,(129)

AΩ(t) = maxω′,ω′′∈Ω

|V (t, ω′)− V (t, ω′′)|.(130)

We will perform a computation similar in the spirit to the discrete system (26), but minding the wider rangeof the flock parameters. So, let us pick a triple ` ∈ (R2n)∗, |`| = 1, ω′, ω′′ ∈ Ω which maximizes AΩ(t),

AΩ(t) = `(V (t, ω′′)− V (t, ω′)).

We abbreviate V (t, ω) = V , V (t, ω′) = V ′, V (t, ω′′) = V ′′, etc. Then by the same argument as in discretecase,

d

dtAΩ(t) = λ

∫R2n

φ(X −X ′′)`(V − V ′′) dµ0(ω) + λ

∫R2n

φ(X −X ′)`(V ′ − V ) dµ0(ω)

≤ λφ(DΩ)

∫R2n

`(V − V ′′ + V ′ − V ) dµ0(ω) = −λMφ(DΩ)AΩ(t).

We also have triviallyd

dtDΩ ≤ AΩ.

So, we recover the same system of ODIs as in the discrete case(27). For general kernels, it simply impliesthat AΩ is a decreasing quantity:

(131) AΩ(t) ≤ AΩ(0)

and hence DΩ(t) . t. In particular, this implies a priori linear bound on the radius of the kinetic flock:x

(132) Suppµt ⊂ BR1+tR2(0).

In the case of heavy tail kernel we obtain the full analogue of Theorem 3.4 in terms of kinetic parameters.

Theorem 5.3 (Alignment for kinetic model). Let µ ∈ Cw∗(R+;M2+(R2n)) be a given solution with compact

initial support and let Ω be a compact domain with Suppµ0 ⊂ Ω. Suppose DΩ is a solution to

(133)

∫ DΩ

DΩ(0)

φ(r) dr =AΩ(0)

λM.


Then the solution µ flocks exponentially fast according to

(134) supt≥0DΩ(t) ≤ DΩ, AΩ(t) ≤ AΩ(0)e−tλMφ(DΩ).

Besides flocking behavior estimates (134) imply that in the fat kernel case the supports of µt remain apriori uniformly bounded

Suppµt ⊂ BR(0), ∀t ≥ 0.

Further refinement of flocking behavior can be provided by establishing estimates on the deformationtensor of the flow-map (X,V ). Straight from the Lagrangian formulation we obtain the system for all t ≥ 0,ω ∈ Ω:

∂t∇X(t, ω) = ∇V (t, ω)

∂t∇V (t, ω) = λ

∫R2n

∇>X(t, ω)∇φ(X(t, ω)−X(t, ω′))⊗ (V (t, ω′)− V (t, ω)) dµ0(ω′)

− λ∇V (t, ω)

∫R2n

φ(X(t, ω)−X(t, ω′)) dµ0(ω′).

(135)

Thus,

(136)d

dt‖∇X‖L∞(Ω) ≤ ‖∇V ‖L∞(Ω),

and again with the use of a functional which maximizes `(∇V (t, ω)), ω ∈ Ω,

(137)d

dt‖∇V ‖L∞(Ω) ≤ λ|∇φ|∞M‖∇X‖L∞(Ω)AΩ(t)− λMφ(DΩ(t))‖∇V ‖L∞(Ω).

For general kernels, we simply know the bound (131) and so the above calculation implies an exponentialbound

(138) ‖∇V ‖L∞(Ω) + ‖∇X‖L∞(Ω) ≤ C1eC2t.

For heavy tail kernels, with the use of (134) we obtain a system of type (33)

d

dt‖∇X‖L∞(Ω) ≤ ‖∇V ‖L∞(Ω)

d

dt‖∇V ‖L∞(Ω) ≤ λ|∇φ|∞M‖∇X‖L∞(Ω)AΩ(0)e−tλMφ(DΩ) − λMφ(DΩ)‖∇V ‖L∞(Ω).

The resulting estimate (34) implies, noting that initially ∇(X0, V0) = Id,

(139) a‖∇X(t)‖2L∞(Ω) + ebt‖∇V (t)‖2L∞(Ω) ≤4√a

b(a+ 1)

where a = λM |∇φ|∞AΩ(0), b = λMφ(DΩ). In particular, ‖∇V (t)‖L∞(Ω) is exponentially decaying.

5.4. Stability. Kantorovich-Rubinstein metric. Contractivity. In this section we provide an analogueof stability property of solutions to (114) similar to one obtained in Section 5.4. On the kinetic level ofdescription analytically suitable way to measure closeness of two flocks is via use of Kantorovich-Rubinsteinmetric (or of Wasserstein-1 distance), which is compatible with weak topology onM+. For two measures ofequal mass µ, ν ∈M1

+, we define

(140) W1(µ, ν) = supLip(g)≤1

∣∣∣∣∫R2n

g(ω) dµ(ω)−∫R2n

g(ω) dν(ω)

∣∣∣∣ .It follows directly from the definition that if Lip(g) ≤ L, then∣∣∣∣∫

R2n

g(ω) dµ(ω)−∫R2n

g(ω) dν(ω)

∣∣∣∣ ≤ LW1(µ, ν).

Lemma 5.4. For a sequence of measures with Suppµn ⊂ BR(0), W1(µn, µ) → 0 if and only if µn → µweakly.

42 ROMAN SHVYDKOY

Proof. Clearly if W1(µn, µ) → 0 then∫R2n g(ω) dµn(ω) →

∫R2n g(ω) dµ(ω) for every Lipschitz function g.

However, Lipschitz functions are dense in C(B2R(0)), hence µn → µ weakly. On the other hand, if µn → µweakly, then µn → µ uniformly on any precompact subset of C(B2R(0)), which, in particular, is the set ofall g with Lip(g) ≤ 1.

So, let us consider two solutions on a common interval of existence µ, ν ∈ Cw∗([0, T );M+) with compactinitial supports which we confine into a fixed compact domain Ω

Suppµ0 ∪ Supp ν0 ⊂ Ω.

We also assume that the solutions have equal masses Mµ = Mν and momenta V µ = V ν . Clearly,

d

dt‖Xµ(t)−Xν(t)‖L∞(Ω) ≤ ‖Vµ(t)− Vν(t)‖L∞(Ω).

For the velocities we apply the same strategy as usual by fixing a maximizing functional ` and computing

d

dt‖Vµ(t)− Vν(t)‖L∞(Ω) ≤ λ

∫R2n

φ(Xµ −X ′µ)`(V ′µ − Vµ) dµ0(ω′)

− λ∫R2n

φ(Xν −X ′ν)`(V ′ν − Vν) dν0(ω′)

= λ

∫R2n

φ(Xµ −X ′µ)`(V ′µ − Vµ)[ dµ0(ω′)− dν0(ω′)]

+ λ

∫R2n

[φ(Xµ −X ′µ)− φ(Xν −X ′ν)]`(V ′µ − Vµ) dν0(ω′)

+ λ

∫R2n

φ(Xν −X ′ν)`((V ′µ − V ′ν)− (Vµ − Vν)) dν0(ω′)

There are three terms to estimate on the right hand side. For the first we use the KR-distance:

λ‖φ‖W 1,∞(‖∇Xµ‖L∞(Ω)Aµ,Ω(t) + ‖∇Vµ‖L∞(Ω)

)W1(µ0, ν0).

The second term in bounded by

2λ|∇φ|∞M‖Xµ(t)−Xν(t)‖L∞(Ω)Aµ,Ω(t).

For the last term we use maximality of `(Vµ − Vν) and pull out the kernel first

λ

∫R2n

φ(Xν −X ′ν)`((V ′µ − V ′ν)− (Vµ − Vν)) dν0(ω′) ≤ λφ(Dν,Ω)

∫R2n

`((V ′µ − V ′ν)− (Vµ − Vν)) dν0(ω′)

= λφ(Dν,Ω)`

[∫R2n

(V ′µ − V ′ν) dν0(ω′)

]− λφ(Dν,Ω)M‖Vµ(t)− Vν(t)‖L∞(Ω)

= λφ(Dν,Ω)`

[∫R2n

V ′µ dν0(ω′)− V ν]− λφ(Dν,Ω)M‖Vµ(t)− Vν(t)‖L∞(Ω)

= λφ(Dν,Ω)`

[∫R2n

V ′µ dν0(ω′)−∫R2n

V ′µ dµ0(ω′)

]− λφ(Dν,Ω)M‖Vµ(t)− Vν(t)‖L∞(Ω)

where in the last step we used equality of momenta. Continuing we obtain

≤ λ|φ|∞‖∇Vµ‖L∞(Ω)W1(µ0, ν0)− λφ(Dν,Ω)M‖Vµ(t)− Vν(t)‖L∞(Ω).

Putting all the estimates together we obtain the system

d

dt‖Xµ(t)−Xν(t)‖L∞(Ω) ≤ ‖Vµ(t)− Vν(t)‖L∞(Ω)

d

dt‖Vµ(t)− Vν(t)‖L∞(Ω) ≤ λ‖φ‖W 1,∞(‖∇Xµ‖L∞(Ω)Aµ,Ω(t) + ‖∇Vµ‖L∞(Ω))W1(µ0, ν0)

+ 2λ|∇φ|∞M‖Xµ(t)−Xν(t)‖L∞(Ω)Aµ,Ω(t)

− λφ(Dν,Ω)M‖Vµ(t)− Vν(t)‖L∞(Ω).


Two conclusions follow as before from the system above. For general kernels φ we can use the bound(138) to estimate

d

dt(‖Xµ(t)−Xν(t)‖L∞(Ω) + ‖Vµ(t)− Vν(t)‖L∞(Ω)) ≤ C1e

C2tW1(µ0, ν0) + C3‖Xµ(t)−Xν(t)‖L∞(Ω).

Since initially ‖Xµ(0)−Xν(0)‖L∞(Ω) + ‖Vµ(0)− Vν(0)‖L∞(Ω) = 0, we obtain

(141) ‖Xµ(t)−Xν(t)‖L∞(Ω) + ‖Vµ(t)− Vν(t)‖L∞(Ω) ≤ CeCtW1(µ0, ν0),

for some C > 0 which depends only on initial condition and the kernel.For heavy tail kernels, we use more robust estimate on the deformation tensor (139) and on the diameter

and amplitude (134) to conclude

d

dt‖Vµ(t)− Vν(t)‖L∞(Ω) ≤ ae−bt[W1(µ0, ν0) + ‖Xµ(t)−Xν(t)‖L∞(Ω)]− b‖Vµ(t)− Vν(t)‖L∞(Ω).

So, we obtain the same system (33) but for the new pair

x = W1(µ0, ν0) + ‖Xµ(t)−Xν(t)‖L∞(Ω), v = ‖Vµ(t)− Vν(t)‖L∞(Ω).

Noting that x(0) = W1(µ0, ν0), and v(0) = 0, we obtain from (34)

(142) ‖Xµ(t)−Xν(t)‖L∞(Ω) ≤ CW1(µ0, ν0), ‖Vµ(t)− Vν(t)‖L∞(Ω) ≤ Ce−ctW1(µ0, ν0),

for all time t > 0, where C, c > 0 depend only on the initial kinetic diameter of the flocks, mass, and thekernel.

Although (141) and (142) by themselves express characteristics stability of the flock, the ultimate appli-cation lies in estimating the KR-distance W1(µt, νt) and establishing contractivity of the kinetic dynamics.So, let us assume that on a given time interval [0, T ) we have two solutions µ, ν ∈ Cw([0, T );M+) withbounded supports Suppµ0 ∪ Supp ν0 ∈ Ω, where Ω is convex and compact, for example we can take convexhull of Suppµ0 ∪ Supp ν0. Let us fix a function h with Lip(h) ≤ 1, and use conservation law (127)∫

R2n

h(ω) dµt −∫R2n

h(ω) dνt =

∫R2n

h(Xµ, Vµ) dµ0 −∫R2n

h(Xν , Vν) dν0

=

∫R2n

h(Xµ, Vµ)( dµ0 − dν0) +

∫R2n

[h(Xµ, Vµ)− h(Xν , Vν)] dν0

≤ LipΩ(h(Xµ, Vµ))W1(µ0, ν0) +M(‖Xµ −Xν‖L∞(Ω) + ‖Vµ − Vν‖L∞(Ω)),

Using that

LipΩ(h(Xµ, Vµ)) ≤ ‖∇Vµ‖L∞(Ω) + ‖∇Xµ‖L∞(Ω),

and applying the stability and deformation estimates (138), (139), (141), (142) we conclude the followingbounds

W1(µt, νt) ≤ CeCtW1(µ0, ν0), general kernels,(143)

W1(µt, νt) ≤ CW1(µ0, ν0), heavy tail kernels.(144)

5.5. Mean-field limit. The mean-field limit refers to passage from discrete to kinetic system as N →∞ inthe scaling regime the range of the interactions remains independent of N , i.e. φ(x, y) remains unrescaled.

The analysis of the previous section makes passing to the limit N → 0 almost trivial. Let us start witha given measure µ0 ∈ M+ with compact support. We discretize it in the classical atomic approximation.We consider a box Q of side length L containing Suppµ0, and decompose it into Nn subboxes of side lengthL/N , denote them Qk, k = 1, . . . , Nn. Next we find a ωk = (vk, xk) such that

vk =1

µ0(Qk)

∫Qk

v dµ0(x, v).

44 ROMAN SHVYDKOY

Suppµ0

Qk

L

L/N

We dismiss those Qk which have no mass. Finally, we define

µN0 =

Nn∑k=1

µ0(Qk)δωk .

Clearly, all µN0 ’s have the same momentum and mass. More-over, µN0 → µ0 weakly, and all supports SuppµN0 are confinedto the same box Q. Let us define the empirical measure

µNt =

Nn∑k=1

µ0(Qk)δωk(t),

where ωk(t) = (xk(t), vk(t)) solves the Cucker-Smale system (4)with masses mk = µ0(Qk). These measures will have uniformlybounded supports on any given time interval [0, T ], and in fact

on all R+ if φ has heavy tail. Hence, estimates (143), (144) apply to give us

W1(µNt , µMt ) ≤ C(T )W1(µN0 , µ

M0 ), ∀t < T,

and

W1(µNt , µMt ) ≤ CW1(µN0 , µ

M0 ), ∀t > 0,

respectively. This means at any time t, µNt is a Cauchy sequence in M+, hence there exists a weak limitµNt → µt, uniform on any [0, T ], and R+, respectively. The following lemma concludes the passage.

Lemma 5.5 (Stability under weak limits). Suppose a sequence of solutions µn ∈ Cw∗([0, T );M+) withSuppµnt ⊂ BR(0), for all t < T and n ∈ N converges weakly pointwise, i.e. µnt → µt for all 0 ≤ t < T . Thenµ ∈ Cw∗([0, T );M+) is a weak solution to (114).

Proof. Weak continuity will follow immediately from (118) once it is established. It is clear that all thelinear terms in (118) converge to the natural limits. As to F (µnt ), note that the family of functions φ(x−·)(· − v)(x,v)∈BR(0) is uniformly Lipschitz on BR(0), hence precompact in C(BR(0)). So, F (µnt )(x, v) →F (µt)(x, v) converges uniformly on BR(0). This implies∫ t

0

∫R2n

F (µns )(x, v) · ∇vg(x, v) dµns (x, v) ds→∫ t

0

∫R2n

F (µs)(x, v) · ∇vg(x, v) dµs(x, v) ds.

Indeed, adding and subtracting cross-terms we obtain trivially∫ t

0

∫R2n

F (µs)(x, v) · ∇vg(x, v)( dµns (x, v)− dµs(x, v)) ds→ 0,

and∣∣∣∣∫ t

0

∫R2n

[F (µns )(x, v)− F (µs)(x, v)] · ∇vg(x, v) dµns (x, v) ds

∣∣∣∣≤ |∇g|∞

∫ t

0

∫R2n

‖F (µns )− F (µs)‖L∞(BR(0)) dµns (x, v) ds

= CMn0

∫ t

0

‖F (µns )− F (µs)‖L∞(BR(0)) ds→ 0.

If the initial measure µ0 is in fact absolutely continuous dµ0(ω) = f0(ω) dω, then the resulting solutionwill give density f(t, ω) obtained by the transport along characteristic flow according to (127). Explicitlywe have the Liouville formula for the Jacobian of the flow:

det∇ω(X,V )(ω, t) = exp

−λn

∫ t

0

φ ∗ ρ(X(ω, s), s) ds

,


where ρ is the macroscopic density measure dρ(x, t) =∫Rn µt(x, v), i.e. first marginal of µt. Note that

φ ∗ ρ ∈ C∞. Then

f(X(ω, t), V (ω, t), t) = f0(ω) exp

λn

∫ t

0

φ ∗ ρ(X(ω, s), s) ds

.

Inverting the flow we recover f(t). It is clear that f(t) inherits smoothness of the initial condition as well,in view of the smoothness of the flow map. At the C1 level it is seen from (138). Further regularity can bededuced by proving higher regularity of the flow map.

5.6. Notes and References. The kinetic alignment model was first derived in Ha and Tadmor [48] andshortly after justified through the mean-field limit by Ha and Liu [46]. Asymptotic flocking based oncharacteristic flow approach was proved in Carrillo et al [14], where other Boltzmann-type kinetic modelswere derived, see [1, 15] for comprehensive surveys. Stability estimates were established in [45]. Kineticversion of control problem considered in [12] was addressed by Piccoli, Rossi, Trelat in [79]. The main resultis existence of a control mechanism with finite time support which steers the kinetic transport into alignment.The corrector method, technically, has never been applied on kinetic level. However, it is likely to lead to asimilar result, see Chapter 7 for hydrodynamic version.

Justification of kinetic formulation for singular models remains a challenging problem even at the stageof defining the field F (µ) unless singularity is integrable. The difficulty is more manageable in the weaklysingular case β < 1

2 , where Mucha and Peszek [73] established mean-field limit and existence results.

46 ROMAN SHVYDKOY

6. Macroscopic description. Hydrodynamic limit.

By taking v-moments of (114) we can read off the system for macroscopic density and momentum

(145)

ρt +∇ · (ρu) = 0,

(ρu)t +∇x · (ρu⊗ u +R) =

∫Rnρ(x)ρ(y)(u(y)− u(x))φ(x− y) dy,

where R is the Reynolds stress tensor

R(x, t) =

∫Rn

(v − u(x, t))⊗ (v − u(x, t))f(x, v, t) dv.

One can formally close the system by considering a monokinetic density ansatz concentrated at the macro-scopic velocity u:

(146) f(x, v, t) = ρ(x, t)δ(v − u(x, t)).

Clearly, such an ansatz removes the stress, R = 0, and hence we obtain a closed system

(147)

ρt +∇ · (ρu) = 0,

(ρu)t +∇x · (ρu⊗ u) =

∫Rnρ(x)ρ(y)(u(y)− u(x))φ(x− y) dy,

One drawback of dealing with the system in conservation form is that the momentum variable ρu does notenjoy the maximum principle, inherent for the alignment dynamics. Moreover, from the point of view ofstudying regularity writing (147) for the momentum-density pair would require division by ρ: ρu ⊗ u =1ρ (ρu)⊗ (ρu), which necessitates no-vacuum assumption on solutions. Instead we will study the system for

velocity-density pair:

(148)

ρt +∇ · (ρu) = 0,

ut + u · ∇u =

∫Rnρ(y)(u(y)− u(x))φ(x− y) dy,

It is called pressureless Euler Alignment System (EAS). Clearly, (147) and (148) are equivalent under novacuum condition, and solutions of (148) are also solutions of (147) for any pair (ρ,u) with or withoutvacuum. Moreover, EAS (148) enjoys maximum principle, and is more amenable to well-posedness analysis,which we will discuss in great detail in coming chapters.

The goal of this chapter is to provide a rigorous derivation of the pressureless Euler Alignment systemfrom the kinetic description. Let us note right away that for a smooth pair (ρ,u) the monokinetic ansatz(146) is a measure-valued solution to the kinetic model (114) in the sense of Definition 5.1 if and only if it isa solution to the EAS (148). However, it is not a satisfactory justification considering that smooth solutionsare more practical in use. One way to force a smooth distribution f to converge to the monokinetic one is toconsider a modified model where solutions are being penalized for deviating from the monokinetic measure.To achieve this, we consider a model with strong local alignment

(149) ∂tfε + v · ∇xfε + λ∇v · [fεF (fε)] +

1

ε∇v[fε(uερε,δ − v)] = 0,

where uερε,δ is a special density-renormalized mollification to be defined precisely in the next section. Wewill justify mean-field limit from the corresponding discrete system for this mollified model before we moveto the main goal of this chapter, which is pass from kinetic (149) to pressureless EAS. Namely, we show inTheorem 6.2 that all solutions to (149) converge to the monokinetic one (146) as long as (ρ,u) solves theEAS (148) in the scaling regime

δ ∼ ε2.

The result will be proved on Rn with bounded support or on the torus.


6.1. Multi-scale model and its justification. Consider the mollifier

(150) ψ(r) =c

(1 + r2)n+γ

2

, γ > 1,

where c is a normalizing constant so that∫Rn ψ(|x|) dx = 1. Denote ψδ = 1

δnψ(r/δ). We fix a density

ρ ∈ L1(Rn) in what follows. For a function g ∈ L∞(Rn) we consider the standard mollification gδ = ψδ ∗ g,and a special density-renormalized mollification defined by

(151) gρ,δ =

((gρ)δρδ

)δ

.

Let us note right away, that ρδ(x) > 0 for all x ∈ Rn due to infinite tails of the mollifier. At the same time,∣∣∣∣ (gρ)δρδ

∣∣∣∣∞≤ |g|∞.

This insures that further mollification is legitimate. The resulting function is locally smooth with a fewremarkable properties which we address next.

First, the density-renormalized mollification is a symmetric operation relative to the ρ-weighted innerproduct:

(152) 〈f, g〉ρ =

∫Rnf(x)g(x)ρ(x) dx.

We have

(153) 〈fρ,δ, g〉ρ = 〈f, gρ,δ〉ρ.Second, if g ∈ W 1,∞, its density-mollification approximates g well, and what is crucial, it does not rely onany regularity of the density itself, just its mass M !

Lemma 6.1. Let g ∈W 1,∞. Then

(154) ‖gρ,δ − g‖L1(ρdx)≤ δCψM‖g‖W 1,∞ .

Proof. We have

‖gρ,δ − g‖L1(ρdx)= sup|f |∞≤1

∫Rnfρ(gρ,δ − g) dx.

So, let us fix |f |∞ ≤ 1, and compute∫Rnfρ(gρ,δ − g) dx =

∫Rnfρ(gρ,δ − gδ) dx+

∫Rnfρ(gδ − g) dx.

In the last integral we use simply |gδ − g|∞ ≤ Cδ‖g‖W 1,∞ , so it is bounded my CMδ‖g‖W 1,∞ . In the firstintegral we switch the outer mollification back on fρ:∫

Rnfρ(gρ,δ − gδ) dx =

∫Rn

(fρ)δ

((gρ)δρδ− g)

dx =

∫Rn

(fρ)δρδ

((gρ)δ − gρδ) dx.

We have

(gρ)δ(x)− g(x)ρδ(x) =

∫Rnψδ(x− y)ρ(y)[g(y)− g(x)] dy ≤ δ‖g‖W 1,∞

∫Rnψδ(x− y)ρ(y) dy,

where ψ(r) = rψ(r), still an integrable kernel. At the same time,

(fρ)δ(x)

ρδ(x)≤ |f |∞ ≤ 1, ∀x ∈ Rn.

So, ∣∣∣∣∫Rnfρ(gρ,δ − gδ) dx

∣∣∣∣ ≤ δ‖g‖W 1,∞

∫R2n

ψδ(x− y)ρ(y) dy dx ≤ CMδ‖g‖W 1,∞ .

48 ROMAN SHVYDKOY

For fixed pair of parameters ε, δ > 0, we consider the model given by

∂tfε + v · ∇xfε + λ∇v · [fεF (fε)] +

1

ε∇v[fε(uε,δ − v)] = 0,

uε,δ : = uερε,δ.(155)

Notably, uε,δ is defined only in terms of the macroscopic momentum ρu and density, which are marginalsof f . So, even if f = µ is a measure, the mollifications (ρu)δ, ρδ are smooth functions, and as stated earlierρδ > 0. Moreover, if µ has a bounded support then

(ρu)δρδ

(x) =

∫R2n ψδ(x− y)w dµ(y, w)∫R2n ψδ(x− y) dµ(y, w)

≤ diam(Suppµ).

This defines a bounded function, which then produces the smooth field uε,δ we used to define the model.This makes the model extendable to the class of measure-valued solutions as in Definition 5.1. In particular,one can verify directly that the empirical measure (116) solves (155) and and only if the discrete variables(xi,vi) satisfy

(156)

xi = vi,

vi = λ

N∑j=1

mjφ(xi − xj)(vj − vi) +1

ε

∫Rnψδ(xi − y)

∑kmkψδ(|y − xk|)(vk − vi)∑

kmkψδ(|y − xk|)dy,

Note that this represents an averaged version of Motsch-Tadmor model which has a restored symmetry –its total momentum is conserved. The same is true of the kinetic model (155). Clearly the system (156) isglobally well-posed.

We now address the mean-field limit procedure. Following the methodology carried out in the classicalcase, we only need to assess the contribution of the new term in (155). The corresponding V -characteristicsreads

d

dtV (ω, t) = λF (µt)(X,V ) +

1

εGδ(X,V )

Gδ(X,V ) =

∫Rnψδ(X − y)

∫R2n ψδ(y −X(ω′, t))[V (ω′, t)− V (ω, t)] dµ0(ω′)∫

R2n ψδ(y −X(ω′, t)) dµ0(ω′)dy

So, we have the maximum principle and consequently the results of Section 5.3 on amplitude and alignmentestimates apply. As far as deformation estimates are concerned, it is clearly not expected that they willremain independent of ε, δ. But so long as ε, δ are fixed the extra terms we deal with still remain smooth.So, we fix a finite time interval [0, T ], and a compact domain Ω containing Suppµ0. With the use of uniformbound on the amplitude AΩ we obtain following our estimates in (135)–(137)

d

dt‖∇V ‖L∞(Ω) ≤ C‖∇X‖L∞(Ω) +

1

ε‖∇Gδ(X,V )‖L∞(Ω),

while differentiating Gδ we obtain

‖∇Gδ(X,V )‖L∞(Ω) .1

δ‖∇X‖L∞(Ω) + ‖∇V ‖L∞(Ω).

So, for the quantity Y = ‖∇X‖L∞(Ω) + ‖∇V ‖L∞(Ω) we obtain a simple Gronwall’s inequality Yt ≤ Cε,δY .Consequently, we have a bound on Y on any finite time interval.

Moving on to stability estimates, we use the same setup as in Section 5.4. We find

d

dt‖Vµ(t)− Vν(t)‖L∞(Ω) ≤ CTW1(µ0, ν0) + C‖Xµ(t)−Xν(t)‖L∞(Ω)

+1

ε‖Gδ(Xµ, Vµ)(t)−Gδ(Xν , Vν)(t)‖L∞(Ω).


It remains to estimate the last term. We denote for short V ′ = V (ω′, t), X ′ = X(ω′, t), etc. We obtain

Gδ(Xµ, Vµ)−Gδ(Xν , Vν) =

∫Rnψδ(Xµ − y)

∫R2n ψδ(y −X ′µ)[V ′µ − Vµ] dµ0∫

R2n ψδ(y −X ′µ) dµ0dy

−∫Rnψδ(Xν − y)

∫R2n ψδ(y −X ′ν)[V ′ν − Vν ] dν0∫

R2n ψδ(y −X ′ν) dν0dy.

Let us break it down further into

J1 + J2 + J3 + J4 + J5 =


∫R2n ψδ(y −X ′µ)[V ′µ − V ′ν + Vν − Vµ] dµ0∫


+


∫R2n ψδ(y −X ′µ)[V ′ν − Vν ][ dµ0 − dν0]∫


+


∫R2n [ψδ(y −X ′µ)− ψδ(y −X ′ν)][V ′ν − Vν ] dν0∫


+

∫Rn

[ψδ(Xµ − y)− ψδ(Xν − y)]

∫R2n ψδ(y −X ′ν)[V ′ν − Vν ] dν0∫


+

∫Rnψδ(Xν − y)

∫R2n

ψδ(y −X ′ν)[V ′ν − Vν ] dν0

[1∫

R2n ψδ(y −X ′µ) dµ0

− 1∫R2n ψδ(y −X ′ν) dν0

]dy.

Let us estimate term by term. The first one is straightforward:

‖J1‖L∞(Ω) ≤ C‖Vµ(t)− Vν(t)‖L∞(Ω).

In the remaining terms we rely on the following observation. Since we work on a finite interval t ≤ T ,all the characteristics have a finite reach ‖Xµ‖L∞(Ω) + ‖Xν‖L∞(Ω) ≤ RT . So, if |y| < 2RT we have Cδ ≥ψδ(y −X(ω′, t)) ≥ cδ for all ω′ ∈ Ω, t < T . Otherwise, if |y| ≥ 2RT , we have

Cψδ(y) ≥ ψδ(y −X(ω′, t)) ≥ cψδ(y).

With this in mind, we estimate the remaining terms in similar fashion, splitting into cases |y| < 2RT and|y| ≥ 2RT . So, for J2 we have for |y| < 2RT∫

R2n

ψδ(y −X ′µ)[V ′ν − Vν ][ dµ0 − dν0] ≤ C(‖∇Xµ‖L∞(Ω) + ‖∇Vν‖L∞(Ω))W1(µ0, ν0) ≤ CW1(µ0, ν0),

and for |y| ≥ 2RT ,∫R2n

ψδ(y −X ′µ)[V ′ν − Vν ][ dµ0 − dν0] ≤ Cδ(ψδ(y) + |∇ψδ(y)|)W1(µ0, ν0).

Hence,

J2 .W1(µ0, ν0)

[∫|y|<2RT

ψδ(Xµ − y)∫R2n ψδ(y −X ′µ) dµ0

dy +

∫|y|≥2RT

ψδ(y) + |∇ψδ(y)|∫R2n ψδ(y) dµ0

ψδ(y) dy

]≤ CW1(µ0, ν0).

For J3, we have for |y| < 2RT ,

|ψδ(y −X ′µ)− ψδ(y −X ′ν)| ≤ ‖Xµ(t)−Xν(t)‖L∞(Ω),

and for |y| ≥ 2RT ,|ψδ(y −X ′µ)− ψδ(y −X ′ν)| ≤ |∇ψδ(y)|‖Xµ(t)−Xν(t)‖L∞(Ω).

Consequently, by the same computation as above

J3 ≤ C‖Xµ(t)−Xν(t)‖L∞(Ω).

Similarly, for J4 we have the exact same difference of kernels in the outer integral. So, the same argumentapplies

J4 ≤ C‖Xµ(t)−Xν(t)‖L∞(Ω).

50 ROMAN SHVYDKOY

And finally,

1∫R2n ψδ(y −X ′µ) dµ0

− 1∫R2n ψδ(y −X ′ν) dν0

=

∫R2n ψδ(y −X ′ν) dν0 −

∫R2n ψδ(y −X ′µ) dµ0∫


∫R2n ψδ(y −X ′ν) dν0

=

∫R2n [ψδ(y −X ′ν)− ψδ(y −X ′µ)] dν0 +

∫R2n ψδ(y −X ′µ)[ dν0 − dµ0]∫


∫R2n ψδ(y −X ′ν) dν0

.

All the terms are similar to those we encountered before, so we obtain the same estimate:

J5 ≤ C‖Xµ(t)−Xν(t)‖L∞(Ω) + CW1(µ0, ν0).

Ultimately, we obtain

‖Gδ(Xµ, Vµ)(t)−Gδ(Xν , Vν)(t)‖L∞(Ω) .W1(µ0, ν0) + ‖Xµ(t)−Xν(t)‖L∞(Ω) + ‖Vµ(t)− Vν(t)‖L∞(Ω).

Then the grand quantity Y = ‖Xµ(t)−Xν(t)‖L∞(Ω) + ‖Vµ(t)− Vν(t)‖L∞(Ω) satisfies

d

dtY .W1(µ0, ν0) + Y.

Given that Y0 = 0, we obtain

Y (t) .W1(µ0, ν0).

With the stability and deformation estimates at hand we conclude as in Section 5.4

W1(µt, νt) .W1(µ0, ν0), t < T.

This establishes convergence of the empirical measures towards a solution to (155). Moreover, if fε0 is asmooth distribution it will remain so for all times.

6.2. Hydrodynamic limit. Kinetic relative entropy. We work in the settings of a finite time interval[0, T ] and all initial conditions fε0 to (149) being bounded to a common ball |ω| ≤ R0. This guarantees,as we noted in the previous sections, that Supp fε(t) ⊂ |ω| . R+ t, t ≤ T . So, we have a common boundon supports of constructed solutions on a given time interval.

Theorem 6.2. Let (ρ,u) be a classical solution to (148) on time interval [0, T ). Let f = ρ(x, t)δv=u(x,t) be

the corresponding monokinetic solution to (114). Suppose fε0 ∈ C1(R2n) is a family of initial conditions for(149) satisfying

(F1) Supp fε0 ⊂ |w| < R0;(F2) W1(fε0 , f0) ≤ ε;

Then in the scaling regime δ = ε2, we have for all t < T

W1(fε(t), f(t)) +W1(ρε(t), ρ(t)) ≤ C√ε.

Remark 6.3. The proof shows convergence for more general parameter δ as long as δ = o(ε). Specifically,

W1(fε(t), f(t)) +W1(ρε(t), ρ(t)) .

√ε+

δ

ε.

We see that δ = ε2 appears to be most optimal: if δ ε2, the kinetic equation becomes over-resolved withoutimprovement on convergence rate of solutions, if δ ε2, the model is under-resolved and the convergencerate slows down.

Letting

Eε =1

2

∫R2n

|v|2fε(x, v) dxdv, Iε2 =1

2

∫R2n

∫R2n

φ(x− y)|w − v|2fε(y, w, t)fε(x, v, t) dy dw dx dv

we have the following energy balance relation

(157)d

dtEε = −Iε2 +

1

ε

∫Rn

|(ρεuε)δ|2ρεδ

dx− 2

εEε.


The key observation is that the local alignment term in the energy law controls inner energies both relativeto the local field uε and to its density-renormalized version uε,δ. Let us first define

uε(x) =

ρεuε(x)

ρε(x), ρε(x) > 0,

0, ρε(x) = 0.

This defines a bounded but possibly non-smooth extension of uε from the support of the density. However,in the course of the proof we will not rely on any regularity of uε itself.

So, we have the following formulas for internal energies:

EεI :=1

2

∫R2n

|v − uε(x)|2fε(x, v) dxdv = Eε − 1

2

∫Rnρε(x)|uε(x)|2 dx

Eε,δI :=1

2

∫R2n

|v − uε,δ(x)|2fε(x, v) dxdv = Eε −∫Rn


dx+1

2

∫Rnρε(x)|uε,δ(x)|2 dx.

The two energies add up to

EεI + Eε,δI = 2Eε −∫Rn


dx+1

2

∫Rnρε(x)|uε,δ(x)|2 dx− 1

2

∫Rnρε(x)|uε(x)|2 dx.

We now show that the last two terms add up to a negative value. Indeed, for any pair (u, ρ), we have byMinkowski’s inequality

|uρ,δ|2 ≤(∣∣∣∣ (ρu)δ

ρδ

∣∣∣∣2)δ

≤(

(ρ|u|2)δρδ

)δ

.

Thus,∫Rnρ(x)|uρ,δ(x)|2 dx ≤

∫Rnρ(x)

((ρ|u|2)δρδ

)δ

(x) dx =

∫Rnρδ(x)

(ρ|u|2)δρδ

(x) dx

=

∫Rn

(ρ|u|2)δ(x) dx =

∫Rnρ(x)|u(x)|2 dx.

We have shown that

2Eε −∫Rn


dx ≥ EεI + Eε,δI .

Consequently, the following energy inequality holds

(158)d

dtEε ≤ −Iε2 −

1

ε[EεI + Eε,δI ].

Integrating in time, we obtain in particular

(159)

∫ T

0

EεI (t) dt ≤ ε.

So, suppose we have a smooth local solution to (148) with compact support of the flock Supp ρ0. InTheorem 8.1 below we demonstrate that such solutions can be obtained, for example, in the class

(u, ρ) ∈ Cw([0, T );Hm × (Hk ∩ L1+)) ∩ Lip([0, T );Hm−1 × (Hk−1 ∩ L1

+)), m ≥ k + 1 >n

2+ 2.

We consider what we call a kinetic relative entropy

ηε(t) =1

2

∫R2n

|v − u(x, t)|2fε(x, v, t) dxdv.

The role of this quantity is to control deviation of the distribution fε from limiting monokinetic ansatz. Ournext goal is to establish smallness of ηε on the entire time interval:

(160) ηε(t) . ε, ∀δ ≤ ε2.

52 ROMAN SHVYDKOY

Before we write the equation for ηε(t), let us note that since all our solutions are smooth, we can treat alldifferential operations as classical. The macroscopic system for ε-solutions reads

(161)

ρεt +∇ · (ρεuε) = 0,

(ρεuε)t +∇x · (ρεuε ⊗ uε +Rε) =

∫Rnρε(x)ρε(y)(uε(y)− uε(x))φ(x− y) dy

+1

ερε(uε,δ − uε).

Again, we recall that even if each individual component in the inertia term may not be smooth, the sum is:

ρεuε ⊗ uε +Rε =

∫Rn

(v ⊗ v)fε(x, v, t) dv.

So, let us break up ηε(t) into three components

ηε(t) = Eε −∫Rnρεuε · u dx+

1

2

∫Rnρε|u|2 dx.

We already worked out the energy flux for Eε above, let us move to the next two terms:

d

dt

∫Rnρεuε · u dx =

∫Rn∂t(ρ

εuε) · u dx+

∫Rnρεuε · ∂tu dx

=

∫Rn

(ρεuε ⊗ uε +Rε) : ∇u dx+1

2

∫R2n

φ(x− y)ρε(x)ρε(y)(uε(y)− uε(x))(u(x)− u(y)) dy dx

+1

ε

∫Rnρε(uε,δ − uε) · u dx−

∫Rnρεuε ⊗ u : ∇u dx

+

∫R2n

φ(x− y)ρε(x)ρ(y)uε(x)(u(y)− u(x)) dy dx.

d

dt

1

2

∫Rnρε|u|2 dx =

∫Rnρεu · ∂tu dx+

1

2

∫Rn∂tρ

ε|u|2 dx

= −∫Rnρεu⊗ u : ∇u dx+

∫R2n

φ(x− y)ρε(x)ρ(y)u(x)(u(y)− u(x)) dy dx

+

∫Rnρεu⊗ uε : ∇u dx.

All the inertia terms add up to

−∫Rnρε(uε − u)⊗ (uε − u) : ∇u dx ≤ |∇u|∞

∫Rnρε|uε − u|2 dx.

What we see on the right hand side is the macroscopic relative entropy which can be estimated by internalenergy and kinetic relative entropy:

(162)

∫Rnρε|uε − u|2 dx =

∫Rn|uε − v + v − u|2fε(x, v, t) dx dv . EεI + ηε.

Next, the Reynolds stress term is estimated by∫RnRε : ∇u dx ≤ |∇u|∞

∫R2n

|v − uε(x, t)|2fε(x, v, t) dx dv ∼ EεI .

As to the local alignment term, we use Lemma 6.1 and symmetry (153),∫Rnρε(uε,δ − uε) · u dx = 〈uε,δ,u〉ρε − 〈uε,u〉ρε = 〈uε,uρε,δ〉ρε − 〈uε,u〉ρε

= 〈uε,uρε,δ − u〉ρε ≤ C|uε|∞δ|∇u|∞ . δ.So,

1

ε

∫Rnρε(uε,δ − uε) · u dx .

δ

ε.

It remains to make estimates on the global alignment terms which is most involved. What helps controlthese terms is in part the dissipation term Iε2 coming from the energy inequality (158), the relative entropyitself ηε, and the KR-distance between densities W1(ρε, ρ), see (140).


First we make a simple observation that Iε2 dominates the corresponding macroscopic enstrophy:

(163) Iε2 ≥1

2

∫R2n

ρε(x)ρε(y)|uε(x)− uε(y)|2φ(x− y) dy dx.

Indeed, expanding the |w − v|2 term in Iε2 we obtain

Iε2 =

∫R3n

|v|2fε(x, v)ρε(y)φ(x− y) dy dxdv −∫R2n

ρε(x)ρε(y)uε(x)uε(y) dx dy.

Expanding the macroscopic enstrophy we obtain

1

2

∫R2n

ρε(x)ρε(y)|uε(x)− uε(y)|2φ(x− y) dy dx =

∫R2n

ρε(x)ρε(y)|uε(x)|2φ(x− y) dy dx

−∫R2n

ρε(x)ρε(y)uε(x)uε(y) dxdy.

However, the total energy density dominates the macroscopic one:∫Rn|v|2fε(x, v) dv =

∫Rn|v − uε(x)|2fε(x, v) dv + ρε(x)|uε(x)|2 ≥ ρε(x)|uε(x)|2.

This proves (163).First let us collect all the alignment terms :

A =

∫R2n

φ(x− y)ρε(x)ρ(y)(u(x)− uε(x))(u(y)− u(x)) dy dx

+1

2

∫R2n

φ(x− y)ρε(x)ρε(y)(uε(x)− uε(y))(u(x)− u(y)) dy dx.

For the second term we use dissipation (163) to partially absorb it:

1

2

∫R2n

φ(x− y)ρε(x)ρε(y)(uε(x)− uε(y))(u(x)− u(y)) dy dx

=1

2

∫R2n

φ(x− y)ρε(x)ρε(y)|uε(x)− uε(y)|2 dy dx

+

∫R2n

φ(x− y)ρε(x)ρε(y)(uε(x)− uε(y))(u(x)− uε(x)) dy dx

So, the total sum of alignments is bounded by

A ≤ Iε2 +

∫R2n

φ(x− y)ρε(x)ρ(y)(u(x)− uε(x))(u(y)− u(x)) dy dx

+

∫R2n

φ(x− y)ρε(x)ρε(y)(uε(x)− uε(y))(u(x)− uε(x)) dy dx

= Iε2 +

∫R2n

φ(x− y)ρε(x)(ρ(y)− ρε(y))(u(x)− uε(x))(u(y)− u(x)) dy dx

+

∫R2n

φ(x− y)ρε(x)ρε(y)(uε(x)− u(x) + u(y)− uε(y))(u(x)− uε(x)) dy dx

The last term is bounded by the macroscopic relative entropy which from (162) is bounded by EεI + ηε. Letus collect the obtained estimates so far:

d

dtηε . EεI + ηε +

δ

ε+

∫R2n

φ(x− y)ρε(x)(ρ(y)− ρε(y))(u(x)− uε(x))(u(y)− u(x)) dy dx.

It remains to estimate the last alignment term. We break it up as follows∫Rnρε(x)(u(x)− uε(x))

[∫Rn

u(y)(ρ(y)− ρε(y))φ(x− y) dy

]dx

−∫Rnρε(x)(u(x)− uε(x))u(x)

[∫Rn

(ρ(y)− ρε(y))φ(x− y) dy

]dx.

54 ROMAN SHVYDKOY

The inner integrals are clearly bounded by a constant multiples of W1(ρε, ρ), and the remaining outer integralby√ηε. So, we obtain

(164)d


δ

ε+W1(ρε, ρ)

√ηε.

Lemma 6.4. We have the following bound for all t < T :

W1(ρε(t), ρ(t)) . ε+

[∫ t

0

ηε(s) ds

]1/2

.

Assuming the lemma holds, we obtain

d


δ

ε+ ε√ηε +

√ηε[∫ t

0

ηε(s) ds

]1/2

≤ EεI + ηε +δ

ε+ ε2 + ηε +

∫ t

0

ηε(s) ds.

Integrating to t, and using (159) we obtain

ηε(t) . ηε(0) + ε+δ

ε+

∫ t

0

ηε(s) ds.

At this point we see that δ ∼ ε2 is the optimal value of parameter δ which gives a rate of ε. Given thatd(fε0 , f0) ≤ ε we have

ηε(0) =1

2

∫R2n

|v − u0(x)|2[fε0 (x, v) dx dv − df0(x, v)] ≤ CW1(fε0 , f0) ≤ ε.

So,

ηε(t) . ε+

∫ t

0

ηε(s) ds.

Gronwall’s lemma establishes that

ηε(t) . ε.

Again, in view of Lemma 6.4 this also shows that

W1(ρε(t), ρ(t)) .√ε.

To establish convergence to monokinetic ansatz we fix g ∈ Lip(R2n), |∇g|∞ ≤ 1, and compute∫R2n

g(x, v)[fε(x, v) dx dv − df(x, v)] =

∫R2n

g(x, v)fε(x, v) dxdv −∫Rng(x,u(x))ρ(x) dx

=

∫R2n

(g(x, v)− g(x,u(x)))fε(x, v) dx dv +

∫Rng(x,u(x))(ρε(x)− ρ(x)) dx

≤∫R2n

|v − u(x)|fε(x, v) dxdv + CW1(ρε(t), ρ(t)) .√ηε +

√ε .√ε.

This proves the result.

Proof of Lemma 6.4. In the course of the proof we will use the two characteristic flow maps (X,V ) cor-responding to monokinetic solution, and (Xε, V ε) corresponding to fε. Note that the monokinetic flow issmooth as it corresponds to a measure-valued solution of the classical kinetic Cucker-Smale equation, seeSection 5.3. The flow (Xε, V ε) is also smooth, however, it is unstable as ε→ 0. So, it is prohibited, nor dowe require, to use any regularity estimates on it.

Let us denote another distribution, which is the push-forward of fε0 under the monokinetic flow

fε = (X,V )#fε0 .


Let us fix g ∈ Lip(Rn), |∇g|∞ ≤ 1 and compute∫Rng(x)[ρε(x)− ρ(x)] dx =

∫R2n

g(x)[fε(x, v) dx dv − df(x, v)]

=

∫R2n

g(x)[fε(x, v)− fε(x, v)] dxdv +

∫R2n

g(x)[fε(x, v) dx dv − df(x, v)]

=

∫R2n

[g(Xε(ω, t))− g(X(ω, t))]fε0 (ω) dω +

∫R2n

g(X(ω, t))[fε0 (ω) dω − df0]

≤ |∇g|∞∫R2n

|Xε(ω, t)−X(ω, t)|fε0 (ω) dω + Cε.

Denote

D =

∫R2n

|Xε(ω, t)−X(ω, t)|fε0 (ω) dω.

Differentiating we obtain

d

dtD ≤

∫R2n

|V ε(ω, t)− V (ω, t)|fε0 (ω) dω ≤∫R2n

|V ε(ω, t)− u(Xε(ω, t), t)|fε0 (ω) dω

+

∫R2n

|u(Xε(ω, t), t)− u(X(ω, t), t)|fε0 (ω) dω +

∫R2n

|u(X(ω, t), t)− V (ω, t)|fε0 (ω) dω

≤∫R2n

|v − u(x, t)|fε(x, v, t) dxdv + |∇u|∞D

+

∫R2n

|u(X(ω, t), t)− V (ω, t)|[fε0 (ω) dω − df0(ω)] +

∫R2n

|u(X(ω, t), t)− V (ω, t)|df0(ω)

.√ηε +D +W1(fε0 , f0) + 0,

the latter due to the monokinetic assumption:∫R2n

|u(X(ω, t), t)− V (ω, t)|df0(ω) =

∫R2n

|u(x, t)− v|df(ω, t) = 0.

We thus obtaind

dtD ≤ √ηε +D + ε.

Since D(0) = 0, the lemma follows by Gronwall’s inequality.

6.3. Notes and References. Hydrodynamic limit in the context of alignment dynamics was first addressedin Kang, Vasseur [54] where partial result was established with CS term missing. Later Figalli and Kang[40] extended the result to the full model considering local alignment with rough macroscopic velocity:

(165) ∂tfε + v · ∇xfε + λ∇v · [fεF (fε)] +

1

ε∇v[fε(uε − v)] = 0.

Both works are carried out in the framework of weak solutions to (165) previously constructed by Karper,Mellet, Trivisa [55] for a much wider class of models. The approach is based upon analysis of the macroscopicrelative entropy which necessitates a study of characteristic flow to the rough field uε. The theorem is provedon the torus for no-vacuum solutions as a result of limitations that come from dealing with macroscopicvalues. Following the same methodology but with the use of kinetic relative entropy allows to bypass thoselimitations, so the result presented here is more general. The use of density-renormalized mollification alsobridges the gap between kinetic and discrete systems as we can justify such kinetic formulation via mean-fieldlimit.

A parallel approach was developed in the trilogy of papers by Karper, Mellet, Trivisa [55, 57, 56]. Onestarts from stochastically forced Cucker-Smale

xi = vi,

vi =

N∑j=1

mjφ(|xi − xj |)(vj − vi) +σ∑N

j=1mjψδ(|xi − xj |)

N∑j=1

mjψδ(|xi − xj |)(vj − vi) +√

2σWi

56 ROMAN SHVYDKOY

which results into kinetic Vlasov-Fokker-Plank model

(166) ft + v · ∇xf +∇v(fF (f)) + σ∇v(f(u− v)) = σ∆vf.

In the strong diffusion and local alignment limit σ →∞ solutions Maxwelialize to

(167) f(x, v, t) = ρ(x, t)e−|u(x,t)−v|2 .

Reading off the system for macroscopic quantities one arrives at an isentropic EAS

(168)

ρt +∇ · (ρu) = 0,

(ρu)t +∇x · (ρu⊗ u) +∇p =

∫Rnρ(x)ρ(y)(u(y)− u(x))φ(x− y) dy

with isothermal pressure law p = ρ. This system received much less attention than its pressureless counter-part, see partial results for smooth [25, 56] and singular [28] models. A distinct feature of pressured dynamicsis that solutions, say on the torus, converge to purely uniform distributions ρ = M , as those present theonly possible outcome for a fully aligned flock.

For singular models, we already commented the very issue of defining F (f) becomes central. For weaklysingular kernels β < n, a valuable insight into hydrodynamic limit was offered by Poyato and Soler [81], wholooked into the model with friction, diffusion, and external forcing

εft + εv · ∇xf +∇xψ · ∇vf +∇v(fFε(f)) = ∆vf +∇v(fv),

where ψ is an external potential, and Fε is defined as before with desingularized kernel

φε(r) =1

(ε2 + r2)β/2.

The hydrodynamic limit as ε→ 0 removes inertia terms and one obtains the following system

ρt +∇ · (ρu) = 0, ∇ψ + u = φ ∗ (ρu)− uφ ∗ ρ.Note that the dynamics here it is driven by external potential, as if ψ = const, then the system has onlytrivial traveling wave solutions u ≡ u, ρ = ρ(x− tu).


7. Euler Alignment System

In this chapter we will focus on analysis of long time behavior of solutions to the pressureless EulerAlignment System

(169)

ρt +∇ · (ρu) = 0,

ut + u · ∇u =

∫Ω

φ(x, y)(u(y)− u(x))ρ(y) dy(x, t) ∈ Ω× R+,

where Ω = Tn or Rn. We derived the convolution class of models from kinetic formulation previously, al-though some parts of the analysis presented here extends to more general communication protocols. Ourparticular emphasis will be on the case of local kernels. An attempt to prove alignment in confined envi-ronment such as torus under natural connectivity assumption will lead us to consider a special, and natural,class of topological models.

Let us start with basic energetics of the system and straightforward extensions of the results obtained inthe discrete case.

7.1. Basic properties. Energy law. For symmetric kernels, just as in the discrete case, the system (169)conserves mass and momentum:

ρu =

∫Ω

ρu dx, M =

∫Ω

ρ(x, t) dx.

This allows to predict the limiting alignment velocity to be u = 1M ρu. For convolution kernels, φ(x, y) =

φ(x− y), the system is also Galilean invariant. If this is the case we can always assume that u = 0.The crucial feature of the alignment term in (169) is its commutator representation given by

(170) Cφ(u, ρ) = Lφ(ρu)− Lφ(ρ)u,

where Lφ can take two different forms, either

(171) Lφ(f)(x) =

∫Ω

φ(x, y)f(y) dy,

more suitable for smooth kernels, or

(172) Lφ(f)(x) =

∫Ω

φ(x, y)(f(y)− f(x)) dy,

suitable for singular kernels. Note that for smooth kernels of convolution type we have Lφf = φ ∗ f .The kinetic energy is given by

E =1

2

∫Ω

ρ|u|2 dx.

If u = 0, the energy becomes a measure of alignment due to its relation to the L2-fluctuation functionalgiven by

E =1

2MV2, V2(t) =

1

2

∫Ω×Ω

|u(x, t)− u(y, t)|2ρ(x, t)ρ(y, t) dxdy.

As in the discrete case we mostly work with the fluctuation functionals to make argument independent ofGalilean invariance of the models. Thus, using just symmetry of the kernel, we obtain the basic energy law

(173)d

dtV2 = −2MI2, I2 =

1

2

∫Ω×Ω

φ(x, y)|u(y)− u(x)|2ρ(y)ρ(x) dy.

Since all the macroscopic quantities are measured relative to mass it is natural to introduce the densitymeasure

dmt = ρ(x, t) dx.

In view of the transport nature of the continuity equation, this measure is transported along the flow of u.Namely, if

d

dtX(α, t, t0) = u(X(α, t, t0), t), t > t0

X(α, t0, t0) = α,

58 ROMAN SHVYDKOY

then dmt is a push-forward of dmt0 under X(·, t, t0):

(174) dmt = X(·, t, t0)# dmt0 .

In other words, for any g,

(175)

∫Rng(X(α, t, t0)) dmt0(α) =

∫Rng(x) dmt(x).

We also denote X(·, t, 0) = X(·, t). In particular, the density support is transported by the flow:

Supp ρ(t) = X(Supp ρ0, t).

Denoting the Lagrangian velocity by

(176) v(α, t) = u(X(α, t), t)

for short, and denoting

vαβ = v(α, t)− v(β, t), φαβ = φ(X(α, t)−X(β, t)),

we can rewrite the velocity equation as

(177)d

dtv(α, t) =

∫Ω

φαβ [v(β, t)− v(α, t)] dm0(β).

The Lp-fluctuations are defined by

Vp(t) =1

p

∫Ω×Ω

|v(β, t)− v(α, t)|p dm0(α, β),

where dm0(α, β) = dm0(α)× dm0(β).We can see that these structures bare close resemblance with the discrete and kinetic settings, and

as a result, in Lagrangian coordinates many computations become very similar too. For example, it is

straightforward to see that all Vp’s are decaying in time and (44) translates into just ddtVp ≤ 0.

Another fundamental property of the system (169) is the maximum principle for each scalar velocitycomponent `(u), ` ∈ (Rn)∗, provided the maxima are achieved, which is course true on periodic domain orif u = 0 and the solution decays at infinity on Rn.

7.2. Hydrodynamic flocking and stability. We assume here that the domain is Rn as the discussioncarries over to periodic settings directly, and φ is of convolution type unless noted otherwise.

Let us consider a flock of finite diameter and a compact domain Ω containing Supp ρ0. We define theflock parameters as follows

DΩ(t) = maxα,β∈Ω

|X(α, t)−X(β, t)|, AΩ(t) = maxα,β∈Ω

|v(α, t)− v(β, t)|.

Just as in kinetic case it is important to consider this more general setup in order to have a capability tocompare flocks and study stability. Since the domain Ω is fixed for all time, one can apply the RademacherLemma 3.5 and carry out the same argument as discrete and kinetic cases. Thus, for general non-negativekernels we have the maximum principle

(178)d

dtAΩ(t) ≤ 0.

By expanding Ω to Rn we also see that the global amplitude is not increasing either:

(179) A(t) = supα,β∈Rn

|v(α, t)− v(β, t)|, A(t) ≤ A(0).

For fat kernels we have the following result whose proof is identical to Theorem 3.4.

Theorem 7.1. The system (169) aligns and flocks exponentially fast provided φ is a non-increasing, every-where positive, and satisfying the heavy tail condition. More precisely, if DΩ solves (24), then we have thefollowing estimates

(180) supt≥0DΩ(t) = DΩ, AΩ(t) ≤ AΩ(0)e−λMφ(DΩ)t.


The corresponding hydrodynamic version for Motsch-Tadmor model (55) carries over to macroscopiccontext ad verbatim. For systems with degenerate communication the corrector method yields the samestatement as in Theorem 3.10(ii), which was proved independent of the number of agents. Here one makesuse of macroscopic quantities throughout:

dαβ = −Xαβ ·vαβ|vαβ |

, G3 =

∫R2n

|vαβ |3ψ(dαβ)χ(|Xαβ |) dm0(α, β), etc.

The Lagrangian formulation of the Euler Alignment system given by

d

dtX(α, t) = v(α, t)(181)

d

dtv(α, t) = λ

∫Rnφ(X(α, t)−X(β, t))[v(β, t)− v(α, t)]ρ0(β) dβ(182)

has almost identical structure to its kinetic counterpart (128). This allows to obtain similar stability andregularity estimates for smooth type communication kernels. We start with estimates on deformation andit will be useful to reproduce them pointwise. So, we have

∂t∇X(α, t) = ∇v(α, t)

∂t∇v(α, t) = λ

∫Rn∇>X(α, t)∇φ(X(α, t)−X(β, t))⊗ (v(β, t)− v(α, t))ρ0(β) dβ

− λ∇v(α, t)

∫Rnφ(X(α, t)−X(β, t))ρ0(β) dβ.

In the case of local kernels, we do not hope for good long time control. So, we simply note that the amplitudeASupp ρ0∪α(t) is bounded by initial condition due to (178), and the kernel is bounded. This implies

(183) ‖∇v(t)‖L∞(Rn) + ‖∇X(t)‖L∞(Rn) ≤ C1eC2t,

For heavy tail kernels, we can deduce better long time estimate on every fixed compact domain Ω containingSupp ρ0. So, let us fix such Ω and deduce as in kinetic case

d

dt‖∇X‖L∞(Ω) ≤ ‖∇v‖L∞(Ω)

d

dt‖∇v‖L∞(Ω) ≤ λ|∇φ|∞‖∇X‖L∞(Ω)AΩ(0)e−tλMφ(DΩ) − λMφ(DΩ)‖∇v‖L∞(Ω).

The resulting estimate (34) implies, noting that initially ∇X = Id, ∇v = ∇u0,

(184) a‖∇X(t)‖2L∞(Ω) + ebt‖∇v(t)‖2L∞(Ω) ≤4√a

b(a+ ‖∇u0‖2L∞(Ω))

where a = λMAΩ(0), b = λMφ(DΩ). In particular, we can see that ‖∇v(t)‖2L∞(Ω) decays exponentially fast.

Suppose now we have two flocks with initial densities ρ′0, ρ′′0 . It turns out the our kinetic result can be

carried out almost ad verbatim in the hydrodynamic setting with the use of Kantorovich-Rubinstein distanceon M+(Rn). So, if Ω is a compact domain enclosing both initial flocks

Supp ρ′0 ∪ Supp ρ′′0 ⊂ Ω,

and the masses and momenta are equal, M ′ = M ′′, v′ = v′′, then we obtain

d

dt‖X ′(t)−X ′′(t)‖L∞(Ω) ≤ ‖v′(t)− v′′(t)‖L∞(Ω)

d

dt‖v′(t)− v′′(t)‖L∞(Ω) ≤ λ‖φ‖W 1,∞(‖∇X ′‖L∞(Ω)A′Ω(t) + ‖∇v′‖L∞(Ω))d(ρ′0, ρ

′′0)

+ 2λ|∇φ|∞M‖X ′(t)−X ′′(t)‖L∞(Ω)A′Ω(t)

− λφ(D′′Ω)M‖v′(t)− v′′(t)‖L∞(Ω).

Subsequently, with the use of (183) and (184), and recalling that characteristics start from the same valuesand ‖v′(0)− v′′(0)‖L∞(Ω) = ‖u′0 − u′′0‖L∞(Ω), we obtain for general kernels

(185) ‖X ′(t)−X ′′(t)‖L∞(Ω) + ‖v′(t)− v′′(t)‖L∞(Ω) ≤ CeCt[W1(ρ′0, ρ′′0) + ‖u′0 − u′′0‖L∞(Ω)]

60 ROMAN SHVYDKOY

Figure 8. Emerging global behavior from local alignment

while for kernels with heavy tail,

‖X ′(t)−X ′′(t)‖L∞(Ω) ≤ C[W1(ρ′0, ρ′′0) + ‖u′0 − u′′0‖L∞(Ω)],(186)

‖v′(t)− v′′(t)‖L∞(Ω) ≤ Ce−ct[W1(ρ′0, ρ′′0) + ‖u′0 − u′′0‖L∞(Ω)].(187)

Continuing with the same KR-distance computation leading to (143) - (144) we obtain

W1(ρ′(t), ρ′′(t)) ≤ CeCt[W1(ρ′0, ρ′′0) + ‖u′0 − u′′0‖L∞(Ω)], general kernels,(188)

W1(ρ′(t), ρ′′(t)) ≤ C[W1(ρ′0, ρ′′0) + ‖u′0 − u′′0‖L∞(Ω)], heavy tail kernels.(189)

Remark 7.2. On the real line R the KR-distance between two densities ρ′, ρ′′ is equal to L1-norm betweenthe corresponding cumulative distribution functions F ′(x) =

∫ x−∞ ρ′(y) dy, F ′′(x) =

∫ x−∞ ρ′′(y) dy:

W1(ρ′, ρ′′) = ‖F ′ − F ′′‖L1 .

See [98] for details.

7.3. Spectral method. Hydrodynamic connectivity. In the next two section we will address thefundamental question: can alignment still occur in systems with purely local interactions? Can we seetransient clustering before global alignment emerges, see Figure 8?

It is clear that the role of connectivity in addressing this question becomes more prominent. We alreadysaw in Section 3.3 that in the open space two disconnected flocks can disperse in different directions withoutever aligning. In a confined environment, such as periodic domain, connectivity can be expressed by a lowerbound on the density as a function of time ρ > c(t). This lower bound quantifies connectivity in a waysimilar to the weighted Fiedler number. In this section we make this connection more precise and applythe spectral method to develop a conditional alignment criterion for singular local kernels of rather generalnature: φ(x, y) = φ(y, x) and satisfying

(190) λ1|x−y|<r0|x− y|n+α

≤ φ(x, y, t) ≤ Λ(t)

|x− y|n+α, 0 < α < 2,

here Λ(t) is simply assumed finite for any t ≥ 0. Since the compactness of embedding Hs → L2 is crucial tothis discussion, as well as a uniform lower bound on the density, which is consistent with finiteness of massonly on a compact environment Ω, we restrict ourselves to the periodic domain Ω = Tn.

A quantitative expression of coercivity of the commutator (170) relies on lower bounds on the density.Precisely how large that lower should be in order to ensure alignment is investigated in the following propo-sition.

Theorem 7.3. Let φ be a symmetric, local, singular kernel satisfying (190) and let (ρ,u) be a global strongsolution to (169). Assume that

(191) C ≥ ρ(t, x) ≥ c√1 + t

, C > c > 0.

Then the solution aligns at an algebraic rate. Namely, there exist η > 0 such that

(192)

∫Td|u(t, x)− u|2ρ(t, x) dx ≤ 1

2M tη.


Proof. We consider the family of eigenvalue problems parameterized by time:

(193)

∫Tnφ(x, y)(u(x)− u(y)) dmt(y) = κ(t)u(x), u ∈ H α

2 .

We seek the minimal eigenvalue, which is of course 0 corresponding to the constant eigenfunction. To removethis trivial solution we restrict it to the time-dependent 1-codimensional subspace

(194) Hα2

0 =

u ∈ H α

2 :

∫Tn

u dmt = 0

.

The key issue here is that Hα2

0 depends on time. We will return to this later. So, we seek the second minimal

eigenvalue of (193) restricted to Hα2

0 , as a solution to the variational problem

(195) κ2(t) = 2M infu∈H

α2

0

∫T2n φ(x, y)|u(y)− u(x)|2ρ(t, y)ρ(t, x) dx dy∫

T2n |u(x)− u(y)|2ρ(t, x)ρ(t, y) dxdy.

In view of (190), and the assumed bounds on the density (191), the upper norm is equivalent for the Hα/2,and the lower to L2, so the existence follows classically by compactness. The number κ2(t) bears completeresemblance with the discrete Fiedler number, discussed in Section 3.3. In terms of this Fiedler number theenergy equation (173) take form

(196)d

dtV2 ≤ −κ2(t)V2.

Consequently,

(197) V2(t) ≤ V2(0) exp

−∫ t

0

κ2(s) ds

.

We will derive now the lower bound κ2(t) ≥ c/(1+t) which clearly implies the statement of the proposition.Using the bounds on the density (191), the mean-zero condition on u, and the lower bound of the kernel(190) we obtain∫

T2n

|u(x)− u(y)|2ρ(t, x)ρ(t, y) dx dy = 2M

∫Tn|u(x)|2ρ(t, x) dx ≤ C‖u‖22,∫

T2n

φ(x, y)|u(y)− u(x)|2ρ(t, y)ρ(t, x) dxdy ≥ c

t

∫|x−y|<r0

|u(x)− u(y)|2|x− y|n+α

dxdy.

(198) κ2(t) ≥ c

tinf

u∈Hα2

0

∫|x−y|<r0

|u(x)−u(y)|2|x−y|n+α dxdy

‖u‖22.

Technically, the infimum still depends on time since the mean-zero condition is. So, the last piece to show isthat this infimum stays bounded away from zero. We argue by contradiction. Suppose there is a sequenceof times tk > 0, and uk ∈ Hα/2 with

∫uk dmtk = 0 such that ‖uk‖2 = 1 and

(199)

∫|x−y|<r0

|uk(x)− uk(y)|2|x− y|n+α

dxdy → 0.

The latter, in particular, implies compactness of the sequence ukk in L2. Hence, up to a subsequence,uk → u strongly in L2 and weakly in Hα/2. By the lower-weak-semi-continuity, and (199), we conclude that‖u‖Hα/2 = 0, and hence u is a constant field, with |u| = 1 due to ‖uk‖2 → ‖u‖2.

At the same time, since∫ρ(tk, x) dx = M , there exists a weak∗ limit of a further subsequence mtk → m,

where m is a positive Radon measure on Td with non-trivial total mass m(Td) = M . We now reach acontradiction if we prove the limit

0 =

∫Tn

uk(x)ρ(tk, x) dx→Mu 6= 0.

To prove the claimed limit note that the assumed uniform upper-bound of the density implies∫Td

uk(x)ρ(tk, x) dx−Mu =

∫Td

(uk(x)− u)ρ(tk, x) dx,

62 ROMAN SHVYDKOY

and the latter is clearly bounded by C‖uk − u‖2 → 0.This proves that κ2(t) ≥ c/t, and the result follows.

In the course of the proof we essentially established a statement analogous to Lemma 3.2 in the discretecase. Let us state it separately.

Lemma 7.4. Let κ2(t) be the weighted Fiedler number defined by (195), and suppose that∫ ∞0

κ2(s) ds =∞.

Then the solution aligns: V2 → 0.

We can see now that unconditional flocking is generally achieved under the lower bound on the density,ρ(t, ·) & (1 + t)−1/2. The difficulty is that this lower bound is too restrictive and is not given a priori forany strong solution. Situation improves considerably for the topological models which yield unconditionalflocking under more accessible assumption on the density. We discuss those next.

7.4. Topological models. Adaptive diffusion. Field study of actual animal formations, such as flocks ofstarlings, see [5, 6, 23, 22, 21, 20], confirmed that interactions in these biological system follow a somewhatdifferent protocol. Each agent is capable to probe a certain fixed number of other agents in its proximityrather than all those within a predefined radius. This nearest neighbor rule defines a neighborhood deter-mined by mass rather than Euclidean distance, see Figure 9. Communication depending on the mass of thecrowd is called topological as opposed to conventional metric one based on Euclidean distance |xi − xj |.

In the context of Cucker-Smale flocking models such topological communication can be defined usinga topological distance which is given by the mass of a communication domain in the intermediate regionbetween two agents Ω(xi,xj). For example,

(200) d(xi,xj) =

∑k:xk∈Ω(xi,xj)

mk

1n

,

where the power 1/n is used to bring volume units back to length. Remarkably, models based on topologicalprotocol exhibit a much more robust flocking behavior than those based on metric protocol, which we willdemonstrate in this section.

We construct topological communication based on the following principles:

1. Every agent xj has a finite range of influence, which is a Euclidean ball of radius r0 centered at xi,denoted B(xi, r0).

2. Agent xi feels the influence of another agent xj according to topological distance between them,d(xi,xj), through a symmetric communication domain Ω(xi,xj). The domain itself is symmetricΩ(xi,xj) = Ω(xj ,xi) and inclusive, xi,xj ∈ ∂Ω(xi,xj).

3. Communication φij is inversely proportional to d(xi,xj) based on the principle that the heavier thecrowd between them the weaker communication reach.

Based on the outlined principles, we make the following choice:

(201) φ(xi,xj) =1

dτ (xi,xj)ψ(|xi − xj |),

where ψ is a non-negative function with support containing the ball of radius r0,

(202) ψ(r) ≥ λ1r<r0 ,and τ > 0 is a parameter. The kernel ψ encodes metric dependencies of the kernel, while τ gauges presenceof topological effects.

The choice of the domain Ω(x,y) can be rather flexible. The subsequent analysis goes through as long asit satisfies two basic requirements (refer to Figure 10): a) the region is a subset of the ball determined by[x,y] as its diameter chord, and two cones of opening < π at vertices x and y, b) Ω(x,y) = Ω(y,x), and theboundary of the region is smooth everywhere except for x,y. For simplicity we also assume that topologicalcommunication is homogeneous and isotropic, i.e. Ω(x,y) is constructed by a shift, rotation, and dilation ofa basic domain Ω(−e1, e1). In order to insure symmetry b) we assume that Ω(−e1, e1) = −Ω(−e1, e1).


Figure 9. Topological versus Metric communication

The analogue of Theorem 3.4 follows along the lines of the proof presented in Section 3.4. We have

(203)

∫ ∞D0

ψ(r) dr >A0

λM1− τn⇒ A(t) ≤ A0e

−tλM1− τn ψ(D).

We are however mostly focused on local interactions. So, this observation applies only in special cases, forinstance when the initial flock is very aggregated D0 r0 and almost aligned A0 λM1− τn .

Note that in the macroscopic limit N →∞, the natural interpretation of the topological distance is givenby

d(x, y) =

[∫Ω(x,y)

ρ(t, z) dz

] 1n

.

In fact in 1D, where

d(x, y) =

∣∣∣∣∫ y

x

ρ(t, z) dz

∣∣∣∣ ,this does define a proper metric. Otherwise, strictly speaking, d is a quasi-metric under no-vacuum condition.

The corresponding hydrodynamic system is given by

(204)

ρt +∇ · (ρu) = 0,

ut + u · ∇xu =

∫Ω

φ(x, y)(u(y)− u(x))ρ(y) dy, φ(x, y) =ψ(|x− y|)dτ (x, y)

.

A proper care has to be given to properly define the singular integral operator Lφf and the commutator Cφis this case. These issues as well as well-posedness of (204) are discussed in [89, 83] and will be omitted hereas they go beyond the scope of this text.

Note that the new kernel incorporates a type of adaptive diffusion which enhances dissipation into thinnerregions and moderates it in thicker regions. As a result we can obtain an improvement upon Proposition 7.3which requires a much weaker lower bound on the density, one that we will be able to actually prove at leastin 1D situation.

Theorem 7.5. Let (ρ,u) be a global smooth solution of the topological model (204) on the torus Tn, withτ ≥ n. Assume that the density satisfies the lower bound

(205) ρ(t, x) ≥ c

1 + t, ∀t > 0, x ∈ Tn.

Then the solution aligns with a logarithmic rate given by

(206) |u(t)− u|∞ ≤c√ln t

.

Remark 7.6. The assumption on the density (205) holds automatically in 1D case, as we will see in Section 9,see (312).

Proof. We aim to prove (206) for each component of velocity ui. Let us denote u = ui for short. By theGalilean invariance of the system we can add a constant to u if necessary and assume that u(t) > 0. By themaximum principle the extrema of u(t), denoted u+(t) and u−(t), are monotone.

64 ROMAN SHVYDKOY

x yΩ(x,y)

Figure 10. Communication domains between agents

Step 1: Alignment near extremes.Denote by x+(t) a point where the maximum of u(t, ·) is achieved, and by x−(t) for minimum. Let us fix

a time-dependent δ(t) > 0 to be specified later. Consider the sets

G+δ (t) = u < u+(t)(1− δ(t)), G−δ (t) = u > u−(t)(1 + δ(t)).

The amount flattening near extreme values will be quantified in terms of conditional expectations of theabove sets relative to the local balls B(x±(t), r0). We denote such expectations by

Et[A|B] =mt(A ∩B)

mt(B).

First, let us show that

(207)

∫ ∞0

Et[G±δ (t)|B(x±(t), r0)] dt <∞.

To this purpose let us compute the equation at (x+(t), t) (with the use of Rademacher’s lemma 3.5)

d

dtu+ =

∫Tnφ(x+, y)(u(y)− u+)ρ(y) dy.

We now use our assumption (202) and the fact that τ ≥ n to obtain the bound

(208)λ1r<r0(|x− y|)

dn(x, y)≤ ψ(x− y)dτ−n(x, y)

dτ (x, y)≤Mτ−nφ(x, y).

Thus, we have

−d

dtu+ =

∫φ(x+, y)(u+ − u(y))ρ(y) dy

&∫B(x+,R0)

1

d(x+, y)(u+(t)− u(y))ρ(y) dy

≥ 1

mt(B(x+, r0))

∫G+δ (t)∩B(x+,r0)

(u+ − u(y))ρ(y) dy (since Ω(x+, y) ⊂ B(x+,0 ))

≥ δ(t)u+

mt(B(x+, r0))

∫G+δ (t)∩B(x+,r0)

ρ(y) dy

= δ(t)u+Et[G+δ (t)|B(x+, r0)].

Integrating we obtain∫ ∞0

δ(t)Et[G+δ (t)|B(x+(t), r0)] dt . ln

u+(0)

limt→∞ u+(t)≤ ln

u+(0)

u−(0).


b

b

b

x∗x

x′

Ω(x, x′)

B(x∗, c0r)B(x∗, r)

Figure 11. Ω(x, x′) is trapped in the outer ball if x is close to the center x∗.

Step 2: Use of Campanato-Morrey norm. On this step we show that u does not deviate much fromits averages over local balls, whereby essentially establishing local alignment as depicted in Figure 8. Weexpress such measure of deviation in terms of a Campanato-Morrey metric to be specified later. Let usdenote

ux,r =1

mt(B(x, r))

∫B(x,r)

u(t, z) dmt(z).

Recall that the communication domains Ω(x, x′) are confined to cones intersected with the diametric ball.As a result the following geometric observation is true, see Figure 11.

Claim 7.7. There exists a c0 > 0 depending only on the opening angle of the cones such that for any r > 0and any triple of points (x, x′, x∗) with |x− x∗| < c0r and |x′ − x∗| < r, we have Ω(x, x′) ⊂ B(x∗, r).

Let us fix an arbitrary x∗ ∈ Tn. By Holder inequality, we have the following estimate for any r < r0/2:∫|x−x∗|<c0r

|u(x)− ux∗,r|2ρ(x) dx ≤∫|x−x∗|<c0r|x′−x∗|<r

1

mt(B(x∗, r))|u(x)− u(x′)|2ρ(x)ρ(x′) dx′ dx

using that mt(B(x∗, r)) ≥ mt(Ω(x, x′)) = dn(x, x′)

≤∫|x−x′|<(1+c0)r

1

dn(x, x′)|u(x)− u(x′)|2ρ(x)ρ(x′) dx′ dx

≤ C∫T2n

φ(x, x′)|u(x)− u(x′)|2ρ(x)ρ(x′) dx′ dx.

From the energy equation (173), the right hand side is globally integrable on R+. Consequently, we obtaina global time integrability for the following Campanato-Morrey semi-norm,

(209)

∫ ∞0

[u]2r0 dt <∞, [u]2r0 := supx∗∈Tn,r< r0

2

∫|x−x∗|<c0r

|u(x)− ux∗,r|2ρ(x) dx.

In combination with (207) we obtain

I =

∫ ∞0

(δ(t)Et[G±δ (t)|B(x±(t), r0)] + [u(t)]2r0

)dt <∞.

Denoting A = e2I we have ∫ TA

T

dt

t ln t= 2I for all T > 0.

66 ROMAN SHVYDKOY

Consequently, for any T > 0 there exists a t ∈ [T, TA] such that

[u(t)]2r0 <1

t ln t

Et[G+δ (t)|B(x+(t), r0)] + Et[G−δ (t)|B(x−(t), r0)] <

1

δ(t)t ln t

(210)

By virtue of the lower bound on the density (205) this implies that

(211) supx∗, r<

r02

∫|x−x∗|<c0r

|u(x)− ux∗,r|2 dx ≤ 1

ln t.

Step 3: Sliding averages. Let t ∈ [T, TA] be the moment of time fixed above. Let us fix r = 14r0, which

lies within the reach of the Campanato metric. We will connect the two averages ux+,r and ux−,r slidingalong the line connecting x+ and x−, and show that the fluctuation of those averages is small.

To this end, consider the direction vector n = x+−x−|x+−x−| and define a sequence of balls, Bk = B(xk, c0r), k =

0, . . . ,K, with centers given by x0 = x− and defined recursively by xk+1 = xk + c0rn up to k = K − 1 andending with xK = x+. The point is that the balls overlap significantly: |Bk ∩Bk+1| ≥ c1rn0 .

By the Chebychev inequality, followed by (211) applied to the ball centered at x0, yields

|x ∈ B0 ∩B1 : |u(x)− ux0,r| > η| ≤ 1

η2

∫B0

|u(x)− ux0,r|2 dx ≤ 1

η2 ln t.

Let us set η =2√

c1rn0 ln tso that

|x ∈ B0 ∩B1 : |u(x)− ux0,r| > η| ≤ 1

4|B0 ∩B1|.

By the same argument applied to the fluctuation around the averaged value ux1,r we obtain

|x ∈ B0 ∩B1 : |u(x)− ux1,r| > η| ≤ 1

4|B0 ∩B1|.

Hence, the complements of the two sets must have a common point in the intersection B0 ∩B1:

x ∈ B0 ∩B1 : |u(x)− ux0,r| ≤ η ∩ x ∈ B0 ∩B1 : |u(x)− ux1,r| ≤ η 6= ∅.This implies

|ux0,r − ux1,r| ≤ 2η.

Continuing in similar fashion we recover the same bound for all consecutive pairs of averages:

|uxk,r − uxk+1,r| ≤ 2η.

Hence,

(212) |ux−,r − ux+,r| ≤ 2Kη .1√ln t

.

Notice the absolute bound on K . 1/r0. Furthermore, in view of (210), the follows estimate holds

ux+,r ≥1

mt(B(x+, r))

∫B(x+,r)\G+

δ

u+(t)(1− δ(t)) dmt

≥ u+(t)(1− δ(t))(1− Et[G+δ (t)|B(x+(t), R0)]) ≥ u+(t)(1− δ(t))

(1− 1

δ(t)t ln t

).

Consequently,

u+(t)− ux+,r(t) . δ(t) +1

δ(t)t ln t.

1√t ln t

provided we make the following choice of δ

δ(t) =1√t ln t

.


Exact same argument can be make to estimate the bottom average. In combination with (212) theseimply

|u+(t)− u−(t)| . 1√ln t

.

To conclude we notice that the maximum principle implies

|u+(TA)− u−(TA)| . 1√ln t∼ 1√

ln(TA).

Since T is arbitrary the proof is complete.

Remark 7.8. Theorem 7.5 allows for an extension which improves upon the rate of alignment under morerestrictive bound from below on the density. Specifically, the following statement can be proved along thelines of the above argument: suppose

(213) ρ(t, x) ≥ c

(1 + t)γ, 0 ≤ γ ≤ 1,

then the solution aligns with the following algebraic rate

(214) ‖u(t)− u‖∞ =o(1)

t12 (1−γ)

.

7.5. Notes and References. The pressureless Euler alignment system was derived shortly after introduc-tion of Cucker-Smale model by Ha and Tadmor [48] who also established energy-based flocking behavioranalogous to Cucker, Smale and Ha, Liu results [33, 46]. Theorem 7.1 appeared in [94] almost in its presentform and the stability result is technically new, however is a direct consequence of the analysis presented inthe kinetic context.

The topological model discussed here was introduced in Shvydkoy, Tadmor [89] where Theorem 7.5 andTheorem 7.3 were proved. A prior prototypical model in the context of Cucker-Smale dynamics was proposedby Haskovec in [49]. In essence, the protocol involves mass around the agent d(x, y) = m(B|x−y|(x)).This metric is embedded into classical kernel (3), which makes it non-symmetric. Nonetheless alignmentis established under an infinitely recurring in time connectivity assumption. A similar mass-distance wasexploited in kinetic rank-based models by Blanchet and Degond in [7, 8], see also [53]. Mixed metric-topological models appeared in [74, 84]. We remark that the metric component in our model also plays acrucial role in establishing a priori bounds on the density, [89]. For further discussion see Section 9.5.

68 ROMAN SHVYDKOY

8. Local well-posedness and continuation criteria

The regularity theory for models with smooth communication is understandably quite different fromsingular models – the former is essentially Burgers’ equation with a dumping mechanism, while the latteris a degenerate fractional parabolic system with dissipation in the momentum equation. In this section wewill go through a tedious but necessary for future exposition first step – proving local existence of classicalsolutions. We are not seeking the sharpest spaces to keep exposition simple. However, we do require ourlocal solutions to have certain level of regularity for various phenomena to remain classically verifiable, suchas mass conservation, and existence of characteristic flow. So, the results presented below will respect suchrequirements. We will also limit ourselves to the metric models, where the estimates are not excessivelycontaminated with density dependent coefficients.

8.1. Smooth models. Let us assume throughout that φ is sufficiently smooth to take as many derivativesas necessary in the course of our arguments below. We also assume that φ = φ(x − y) is of convolutiontype and that the environment domain is Rn. The exact same results will carry over to Tn with slightmodifications.

Using the commutator structure of the alignment term and (171) we write the system (169) as

(215)

ρt +∇ · (ρu) = 0,

ut + u · ∇u = φ ∗ (ρu)− u φ ∗ ρ.

Suppose we would like to prove local existence of solutions in Sobolev class u ∈ Hm, ρ ∈ Hk ∩L1+, where L1

+

denotes the set of non-negative functions in L1. Note that the Sobolev embedding does not guarantee thatρ ∈ L1

+ if it is in a higher Sobolev class, yet u ∈ L∞ automatically if m > n2 . Both conditions are natural

quantities to include in the class as they are controlled by dynamics a priori.One can obtain local existence rather easily for a viscous regularization:

(216)

ρt +∇ · (ρu) = ε∆ρ,

ut + u · ∇u = φ ∗ (uρ)− u φ ∗ ρ+ ε∆u.

Indeed, we are going to denote the grand quantity Z = (u, ρ) and consider the equivalent mild formulationof (215):

Z(t) = eεt∆Z0 +

∫ t

0

eε(t−s)∆N (Z(s)) ds,

where N (Z) denotes all the non-linear terms in (215). the argument goes by the standard contractivityargument. Let us fix Z0 ∈ Hm × (Hk ∩ L1

+) and consider the map

T [Z](t) = eεt∆Z0 +

∫ t

0

eε(t−s)∆N (Z(s)) ds.

We need to show that for some small T this maps is a contraction on C([0, T );B1(Z0)), where B1 is under-stood in metric of X = Hm× (Hk ∩L1). Let us invariance, while contractivity following similarly. First, bycontinuity of the heat semigroup,

‖Z0 − eεt∆Z0‖X ≤1

2,

for small t. To estimate the integral we first recall analyticity property of the heat semigroup:

‖∇eεt∆f‖Lp .1√εt‖f‖Lp , 1 ≤ p ≤ ∞.

So, considering the ρ-component we obtain∥∥∥∥∫ t

0

eε(t−s)∆∇ · (ρu) ds

∥∥∥∥L1

≤∫ t

0

1√ε(t− s)

‖uρ(s)‖L1 ds

≤ t1/2

ε1/2sups‖u(s)‖∞‖ρ(s)‖1 ≤

T 1/2

ε1/2(‖Z0‖2 + 1) <

1

2,


for small T . For Hk we argue similarly for L2. So, let ∂k be a multi-index partial derivative of order k. Wewill use the product estimate

‖∂k(uρ)(s)‖L2 ≤ ‖u‖∞‖ρ‖Hk + ‖u‖Hk‖ρ‖∞.It is clear at this point that in order to close the estimates in X we need to assume that n

2 < k ≤ m, inwhich case

‖∂k(uρ)(s)‖L2 ≤ ‖Z(s)‖2X .So, by the same argument we obtain∥∥∥∥∂k ∫ t

0

eε(t−s)∆∇ · (ρu) ds

∥∥∥∥L2

≤∫ t

0

1√ε(t− s)

‖∂k(uρ)(s)‖L2 ds <1

2.

On the velocity side there are two terms to handle. For the transport part, the L2-estimate is straightforward,and∥∥∥∥∂m ∫ t

0

eε(t−s)∆u · ∇u ds

∥∥∥∥L2

≤∫ t

0

1√ε(t− s)

‖∂m−1(u · ∇u)(s)‖L2 ds

≤∫ t

0

1√ε(t− s)

(‖u‖Hm−1‖∇u‖∞ + ‖u‖Hm‖u‖∞) ds <1

4,

provided m > n2 + 1 to ensure embedding of W 1,∞ into Hm. The commutator term φ ∗ (uρ) − u φ ∗ ρ is

even easier, since the derivatives are absorbed by the kernel except when all fall on u, which results in thesame estimate.

We have shown that T : C([0, T );B1(Z0))→ C([0, T );B1(Z0)) is a contraction, and so, we obtain a localsolution on a time interval dependent on ε. Denoting T ∗ the maximal time of existence in C([0, T );X) weshow that T ∗ depends only on the X-norm of the initial condition. We do it by establishing a priori estimatesthat are independent of ε and which will allow us to pass to the limit of vanishing viscosity. So, the grandquantity we are trying to control is

Ym,k = ‖u‖2Hm + ‖ρ‖2Hk + |ρ|21.To start, we write the continuity equation as

ρt + u · ∇ρ+ (∇ · u)ρ = 0.

So, testing with ∂2kρ we obtain

d

dt‖ρ‖2

Hk=

∫(∇ · u)|∂kρ|2 dx−

∫(∂k(u · ∇ρ)− u · ∇∂kρ)∂kρdx−

∫∂k((∇ · u)ρ)∂kρdx− ε‖ρ‖2Hk+1 .

We dismiss the last term. Recalling the classical commutator estimate

(217) ‖∂k(fg)− f∂kg‖2 ≤ |∇f |∞‖g‖Hk−1 + ‖f‖Hk |g|∞,we obtain

d

dt‖ρ‖2

Hk≤ |∇u|∞‖ρ‖2Hk + ‖u‖Hk‖ρ‖Hk |∇ρ|∞ + ‖u‖Hk+1‖ρ‖Hk |ρ|∞

≤ C(|∇u|∞ + |∇ρ|∞ + |ρ|∞)Ym,k,

provided m ≥ k + 1. The L2 norm of ρ obeys a similar estimate trivially, and the L1-norm is conserved.For the velocity equation we apply the same commutator estimate for the material derivative part:∫∂m(u · ∇u)∂mu dx = −

∫∇ · u|∂mu|2 dx +

∫[∂m(u · ∇u) − (u · ∇∂mu)]∂mu dx . |∇u|∞‖u‖2Hm .

For the alignment term, we can put all the derivatives onto the kernel whenever possible and the only termthat is left out is |∂mu|2 φ ∗ ρ with φ ∗ ρ clearly bounded by |φ|∞M – an a priori conserved quantity. So, weobtain

d

dt‖u‖2

Hm≤ (|∇u|∞ + C(|φ|Cm ,M))‖u‖2

Hm.

70 ROMAN SHVYDKOY

The similar bound for ddt‖u‖22 is derived trivially. So, we obtain

(218)d

dt‖u‖2Hm ≤ (|∇u|∞ + C(|φ|Cm ,M))‖u‖2Hm .

It is important to note that this bound is independent of the higher norms of the density. Combining thetwo equations we obtain

(219)d

dtYm,k ≤ C(|∇u|∞ + |∇ρ|∞ + |ρ|∞ + C(|φ|Cm ,M))Ym,k.

Of course |∇u|∞+ |∇ρ|∞+ |ρ|∞ ≤ Ym,k provided k > n2 +1, which adds the last restriction on the exponents

for the argument to work. So, if m ≥ k + 1 > n2 + 2, then

d

dtYm,k ≤ C1Ym,k + C2Y

2m,k.

Solving the Riccati equation gives a uniform bound on the X-norm on a time interval inversely proportionalto ‖Z0‖X , but independent of ε. Thus, solutions to (216) with the same initial data exist on a common timeinterval [0, T0] where they are uniformly bounded in C([0, T0];X).

Let us also note that keeping the dissipative terms in the estimates above also shows that

ε

∫ T0

0

(‖ρ(s)‖2Hk+1 + ‖u(s)‖2Hm+1) ds < C,

where C is independent of ε. Then

‖Zt‖L2 ≤ ‖Z‖2X + ε‖Z‖H2 ≤ ‖Z‖2X + ε‖Z‖Hk+1×Hm+1 .

So, Zt ∈ L2([0, T0];L2). Passing to a subsequence we find a weak limit Zε → Z in L∞([0, T0];X) and(Zε)t → Zt in L2([0, T0];L2) (technically, a limit in L1 may end up being a measure of bounded variation,however as a member of Hk it is absolutely continuous, hence in L1). Since Zt ∈ L2([0, T0];L2), Z isweakly continuous with values in L2. Since L2 is dense in H−m and H−k this implies weak continuityZ ∈ Cw([0, T0];Hm ×Hk). Strong continuity of the density follows from the equations itself:

‖ρt‖L1 ≤ ‖ρ∇u‖1 + ‖u∇ρ‖1 ≤ ‖Z‖2X < C.

Further regularity in time Zt follows from measuring smoothness of the system one level down and performingsimilar product estimates as above.

Having established local existence in X let us come back to (219) and notice that this solution can in factbe extended beyond T0 if we know that

(220)

∫ T0

0

|∇u|∞ dt <∞.

Indeed, |ρ|∞ can be bounded by solving the continuity equation along characteristics

(221) ρ(X(t, α), t) = ρ(α, 0) exp

−∫ t

0

∇ · u(X(s, α), s) ds

.

Using (218) we see that ‖u‖2Hm is also bounded uniformly, and hence so is |∇2u|∞ since m > n2 + 2.

Bootstrapping further by differentiating the continuity equation we bound |∇ρ|∞ in a similar fashion.Having this continuation criterion at hand we can further improve the local existence result by establishing

control over |∇u|∞ directly. First, by the maximum principle, |u(t)|∞ ≤ |u0|∞. Writing equation for onecomponent ∂iuj we have

(222) ∂t∂iuj + u · ∇∂iuj + ∂iu · ∇uj = (∂iφ) ∗ (ujρ)− ∂iujφ ∗ ρ− uj(∂iφ) ∗ ρ.Evaluating at the maximum and minimum and adding up over i, j we obtain

d

dt|∇u|∞ ≤ |∇u|2∞ + CM |u|∞ +M |∇u|∞.

Hence, |∇u|∞ is uniformly bounded a priori on a time interval depending only on |∇u0|−1∞ . So, the contin-

uation criterion allows to extend our local solution to that time interval.Using the transport structure of the momentum equation we can further relax (220) to a condition on

divergence of u, which will be extremely useful in the future. Let us recall that we have uniform bounds on


the deformation tensor of the characteristic flow map given by (183). To translate this to a bound on |∇u|∞in Eulerian coordinates we invert the flow map

∇u(x, t) = ∇−>X(X−1(x, t), t)∇v(X−1(x, t), t).

Using that

|∇−>X(α, t)| ≤ C1

|det∇X(α, t)|eC2t

we recall the Liouville formula for the Jacobian

det∇X(α, t) = exp

∫ t

0

∇ · u(X(α, t), t) dt

.

So, as long as

(223)

∫ T0

0

infx∈Rn

∇ · u(t, x) dt > −∞,

this guarantees that the Jacobian does not vanish. This establishes uniform bound on |∇u(t)|∞ on timeinterval [0, T0).

Let us record the obtained results in the following theorem.

Theorem 8.1 (Local existence of classical solutions). Suppose m ≥ k + 1 > n2 + 2, and (u0, ρ0) ∈ Hm ×

(Hk ∩ L1+). Then there exists time T0 = T0(|∇u0|−1

∞ ,M) and a unique solution to (215) on time interval[0, T0) in the class

(224) (u, ρ) ∈ Cw([0, T0);Hm × (Hk ∩ L1+)) ∩ Lip([0, T0);Hm−1 × (Hk−1 ∩ L1

+))

satisfying the given initial condition. Moreover, any classical local solution on [0, T0) in class (224) andsatisfying (223) can be extended beyond T0.

Theorem 8.1 can be used as a stepping stone to obtain solutions with less smoothness, especially forρ, as long as regularity of u permits to define smooth characteristic flow map with sufficient compactnessproperties.

So, let us first assume u0 ∈ Hm, m > n2 + 1 and ρ0 ∈ L1

+, the most basic assumption on the density.Mollifying the data ((u0)ε, (ρ0)ε), due to Theorem 8.1, we obtain a family of local solutions on a commontime interval T0, since Hm ⊂ W 1,∞, and |∇(u0)ε|∞ ≤ |∇u0|∞. We also note that the estimate (218)holds for any integer m, hence uε ∈ L∞([0, T0);Hm) uniformly. We also have ∂tuε ∈ L∞([0, T0);Hm−1) ⊂L∞([0, T0);L∞). Let us note that Hm ⊂ W 1+δ,∞ for some δ > 0, and the embedding is compact on anybounded set, and of course W 1+δ,∞ ⊂ L∞. So, the Aubin-Lions-Simon Lemma implies that the familyis compact in C([0, T0);W 1+δ,∞) on any bounded set. Passing to a subsequence we obtain a weak limitu in L∞([0, T0);Hm) and strong in C([0, T0);W 1+δ,∞) on any bounded set. The velocity also belongs toCw([0, T0);Hm) as a consequence of the two memberships. Similarly, the family of flows Xε belongs toL∞([0, T0);W 1,∞) ∩ Lip([0, T0);L∞) so is compact in C([0, T0);Cδ), δ < 1, on any bounded set. We canthus claim strong uniform convergence of the flow maps as well. Considering the solution to the continuityequation

(225) ρε(Xε(t, α), t) = ρε(α, 0) exp

−∫ t

0

∇ · uε(Xε(s, α), s) ds

,

we can clearly pass to the strong limit in L1 and the limit satisfies (221).Note that the density essentially plays the role of a passive scalar. In particular, if ρ0 ∈ L1

+∩L∞ initially,then by formula (221) it will remain in the same class on the entire time interval.

The argument above can be elevated to any higher smoothness Hm×(L1+∩W k,∞) as long as m > n

2 +k+1.It goes by differentiating the continuity equation k times and running the same compactness procedure. Weakcontinuity of the density in L1

+∩W k,∞ follows from the established regularity properties of the velocity field

and the corresponding formula for the solution of ∂kρ. The continuation criterion (223) remains valid in thissetting as well.

72 ROMAN SHVYDKOY

Theorem 8.2 (Local existence of strong solutions). Suppose m > n2 + k + 1, k = 0, 1, ... and (u0, ρ0) ∈

Hm × (L1+ ∩W k,∞). Then there exists time T0 = T0(|∇u0|−1

∞ ,M) and a unique solution to (215) on timeinterval [0, T0) in the class

(226) (u, ρ) ∈ Cw([0, T0);Hm × (L1+ ∩W k,∞)).

Moreover, any such solution satisfying (223) can be extended beyond T0.

Lastly, let us discuss a small initial data result on periodic domain Tn, which in fact follows from theprevious computations. Specifically, let us look back at equation (222) and evaluate it at the maximum foreach coordinates, and note that

C∂iφ(uj , ρ) ≤ A(t)|∇φ|∞M,

where A is the global amplitude of the velocity field. Now, on compact domain Tn, we have inf φ = c0 > 0,so

−∂iujφ ∗ ρ ≤ −c0M |∇u|∞,at a point of maximum. This implies

d

dt|∇u|∞ ≤ |∇u|2∞ +A(t)|∇φ|∞M − c0M |∇u|∞.

So, if initially A0 < ε2 and |∇u0|∞ < ε, then A(t) < ε2 for all times and also for some short period of time[0, t∗) we still have |∇u(t)|∞ < 2ε. Let t∗ denotes the maximal such time on the local interval of existence[0, T ). Then for t < t∗, and provided

d

dt|∇u|∞ ≤ ε2|∇φ|∞M − (c0M − ε)|∇u|∞.

Integrating we obtain

|∇u|∞ < ε+ε2|∇φ|∞Mc0M − ε

.

Provided ε < c0M|∇φ|∞M+1 we have |∇u|∞ < 2ε. This implies that t∗ = T , and by continuation criterion (223)

T =∞.

Theorem 8.3 (Small initial data). Suppose (u0, ρ0) is in the class of spaces as stated in either Theorem 8.1or Theorem 8.2 on the periodic domain Tn. Suppose that

(227) A0 < ε2, |∇u0|∞ < ε, ε <c0M

|∇φ|∞M + 1.

Then there exists a global in time solution in the same class for which one has

|∇u(t)|∞ < 2ε, t > 0.

Let us note that on the open space this theorem would not hold. Indeed, we will see in Section 9 thatif the quantity ∂xu0 + φ ∗ ρ0 is negative at least at one point, then the solution blows up. So, no matterhow small initial slope is, one can always arrange a data with compact support of the density such that forsome remote x we have ∂xu0(x) = −ε, yet φ ∗ ρ0(x) < 1

2ε. This solution will blow up in vacuum. However,one can still construct a proper theory of regular solutions in Lagrangian coordinates on the initial domainΩ0 = Supp ρ0 in Sobolev classes (v, ρ) ∈ Hm(Ω0)×Hm−1(Ω0). See Ha, Kang, Kwon [44] for details.

8.2. Singular models. For models with singular kernels the operator Lφ given by (172) becomes thefractional Laplacian. In view of no-vacuum condition necessary to develop a well-posedness theory weconsider the periodic domain, where a uniform lower bound on the density is compatible with finite mass.On Tn the kernel is given by

φ(z) =∑k∈Zn

1

|z + 2πk|n+α, 0 < α < 2,

and we also consider local communications with a smooth cut-off, φ := φ(r)h(r). The important fact is thatwe have a coercivity bound for Lφ:

(228) c1‖f‖Hα − c2‖f‖2 ≤ ‖Lφf‖2 ≤ c3‖f‖Hα + c4‖f‖2.Necessity of the no-vacuum condition can be easily seen by the following example in 1D. Let us consider

a local kernel for simplicity, Suppφ ⊂ B1(0). Let initial density be confined to small ball, Supp ρ0 ⊂ Bε(0),


while u0 = 1 on B10(0), v0 = 0 on B10+ε(0) and smooth in between. Then the density will remain in B2(0)for a time period of at least t < 1, due to u ≤ 1. During this time the momentum equation will remain pureBurgers, hence the solution will evolve into a shock at a time t ∼ ε < 1.

A more subtle blowup can be constructed even when the density vanishes at one point.

Example 8.4. We will see in Section 9 that in dimension 1 there is an extra conservation law for the quantityux + Lφρ. So, if initially zero, it stays zero for all time. Consider the full fractional Laplacian kernel with

α = 1, then Λρ = −ux, where Λ = −(−∆)1/2. Hence, the whole system reduces to

(229) ρt + (ρHρ)x = 0,

where H is the Hilbert transform. This equation appeared in many difference contexts, see [4] for fullaccount. To obtain a blowup we assume that initially ρ0(x) = ρ0(−x), ρ0(0) = ρ′0(0) = 0. It is easy to seethat these conditions are preserved in time. Then writing the equation for Λρ, we obtain

(Λρ)t −H(ρHρ)xx = 0.

Using the identity H(ρHρ) = 12 (Hρ)2 − 1

2ρ2, we write

(Λρ)t − (Λρ)2 −HρHρxx + ρρxx + (ρx)2 = 0.

Evaluating at 0 we obtain Riccati equation

(Λρ)t = (Λρ)2.

Note that Λρ0(0) > 0. Hence, the solution blows up.

The goal of this section will be to obtain local well-posedness result stated next.

Theorem 8.5 (Local existence of classical solutions). Suppose m > n2 + 1, 0 < α < 2, and

(u0, ρ0) ∈ Hm+1(Tn)×Hm+α(Tn),

and ρ0(x) > 0 for all x ∈ Tn. Then there exists time T0 > 0 and a unique non-vacuous solution to (169) ontime interval [0, T0) in the class

(230) u ∈ Cw([0, T0);Hm+1) ∩ L2([0, T0); Hm+1+α/2), ρ ∈ Cw([0, T0);Hm+α).

Moreover, any such solution satisfying

(231) supt∈[0,T0)

(|∇ρ(t)|∞ + |∇u(t)|∞) <∞

can be extended beyond T0.

Performing energy estimates in the same fashion as for smooth models will inevitably create a derivativeoverload on the density. Instead we consider another ”almost conserved” quantity

(232) e = ∇ · u + Lφρ,which satisfies the equation

(233) et +∇ · (ue) = (∇ · u)2 − Tr(∇u)2.

Let us derive it in general for the sake of completeness. Since φ is a convolution kernel, we have

(234) ∂tLφ +∇ · Lφ(ρu) = 0.

Taking the divergence of the velocity equation, we obtain

(235) ∂t(∇ · u) +∇ · [u · ∇u] = ∇ · Lφ(ρu)−∇ · [uLφρ]

with∇ · [uLφρ] = Lφρ∇ · u + u · ∇Lφρ

and∇ · [u · ∇u] = Tr(∇u)2 + u · ∇(∇ · u).

On one hand, combining (234) and (235), we obtain

(236) ∂te+ Lφρ∇ · u + u · ∇e+ Tr(∇u)2 = 0.

Adding and subtracting now (∇ · u)2 produces (233).

74 ROMAN SHVYDKOY

From the order of terms that enter into the formula for e, it is clear that the natural correspondence inregularity for state variables involved is (u ∈ Hm+1) ∼ (ρ ∈ Hm+α).

Note that in 1D the right hand side vanishes and we have a perfect conservation law. This case will bediscussed at length in Section 9.

The grand quantity to be estimated is

Ym = ‖u‖2Hm+1 + ‖e‖2Hm + |e|∞ + |ρ|1 + |ρ−1|∞,which is equivalent to Ym ∼ ‖u‖2Hm+1 + ‖ρ‖2Hm+α + |ρ−1|∞ in view of (228).

Our strategy will be very similar to the smooth case, where we obtain local solutions via viscous reg-ularization, and prove a continuation criterion via a priori estimates on Ym. We assume throughout thatm > n

2 + 1 and 0 < α < 2.So, let us start with (216) and consider the mild formulation

ρ(t) = eεt∆ρ0 −∫ t

0

eε(t−s)∆∇ · (uρ)(s) ds

u(t) = eεt∆u0 −∫ t

0

eε(t−s)∆u · ∇u(s) ds+

∫ t

0

eε(t−s)∆Cφ(u, ρ)(s) ds.

(237)

Let us denote as before Z = (ρ,u) and by T [Z](t) the right hand side of the mild formulation. In order toapply the standard fixed point argument we have to show that T leaves the set C([0, Tδ,ε);Bδ(Z0)) invariant,where Bδ(Z0) is the ball of radius ε around initial condition Z0, and that it is a contraction. We limitourselves to showing details for invariance as the estimates involved in proving Lipschitzness are similar.

First we assume that ρ has no vacuum: ρ0(x) ≥ c0 > 0. Since the metric we are using for ρ ∈ Hm+α

controls L∞ norm, if δ > 0 is small enough then for any ‖ρ− ρ0‖Hm+α < δ one obtains

(238) ρ(x) >1

2c0.

So, let us assume that Z ∈ C([0, T );Bδ(Z0)). It is clear that ‖eεt∆Z0 − Z0‖ < δ2 provided time t is short

enough. The Z has some bound ‖Z‖ ≤ C. Using that let us estimate the norms under the integrals. First,recall that ‖Λαeεt∆‖L2→L2 . 1

(εt)α/2 . In the case α ≥ 1, we have∥∥∥∥∂mΛα

∫ t

0

eε(t−s)∆∇ · (uρ)(s) ds

∥∥∥∥2

.∫ t

0

1

(t− s)α/2 ‖∂m+1(uρ)(s)‖2 ds

≤∫ t

0

1

(t− s)α/2 ‖u‖Hm+1‖ρ‖Hm+α ds ≤ C2t1−α/2 <δ

2,

provided T = T (δ, ε) is small enough. In the case α < 1, we combine instead one full derivatives with theheat semigroup, and the rest ∂m+α gets applied to uρ, which produces a similar bound.

Moving on to the u-equation, we have∥∥∥∥∂m+1

∫ t

0

eε(t−s)∆u · ∇u(s) ds

∥∥∥∥2

.∫ t

0

1

(t− s)1/2‖∂m(u · ∇u)(s)‖2 ds

≤∫ t

0

1

(t− s)α/2 ‖u‖Hm+1‖u‖Hm ds ≤ C2t1/2 <δ

4.

As to the commutator form, for α ≤ 1 the computation is very similar: we combine one derivative with theheat semigroup and for the rest we use (228):

‖∂mCφ(u, ρ)‖2 . ‖u‖m+α‖ρ‖m+α < C2,

and the rest follows as before. When α > 1 we combine α derivatives with the semigroup, and the restfollows as before.

We have proved that ‖T [Z](t) − Z0‖ < δ, for a short time and hence, T leaves C([0, T (δ, ε));Bδ(Z0))invariant.

Now let us make a priori estimates for viscous solutions independent of ε. Note that the dissipation termsin all the following computations are negative and as such will be ignored.


First, evaluating the continuity equation at a point of minimum x− and denoting ρ− = min ρ we readilyobtain

d

dtρ− = −ρ−∇u + ε∆ρ(x−) ≥ −ρ−|∇u|∞.

Hence,d

dt|ρ−1|∞ ≤ |ρ−1|∞|∇u|∞ ≤ |∇u|∞Ym.

Furthermore,

(239)d

dt|e|∞ ≤ |∇u|∞|e|∞ + |∇u|2∞ ≤ |∇u|∞Ym.

Let us continue with estimates on the e-quantity. We have (dropping integral signs)

d

dt‖e‖2

Hm≤ ∂meu · ∇∂me+ ∂me[∂m(u · ∇e)− u · ∇∂me] + ∂me∂m(e∇ · u) + ∂me[(∇ · u)2 − Tr(∇u)2]

In the first term we integrate by part and estimate

|∂meu · ∇∂me| ≤ ‖e‖2Hm|∇u|∞.

For the next commutator term we use (217)

|∂me[∂m(u · ∇e)− u · ∇∂me]| ≤ ‖e‖2Hm|∇u|∞ + ‖e‖Hm‖u‖Hm |∇e|∞.

Using Gagliardo-Nirenberg inequality we estimate the latter term as

‖e‖Hm‖u‖Hm |∇e|∞ ≤ ‖e‖Hm‖u‖θ1Hm+1|∇u|1−θ1∞ ‖e‖θ2

Hm|e|1−θ2∞ ,

where θ1 = n−2(m−1)n−2m and θ2 = 2

2m−n . The two exponents add up to 1, so by the generalized Younginequality,

≤ (‖e‖2Hm

+ ‖u‖2Hm+1)(|e|∞ + |∇u|∞) ≤ (|e|∞ + |∇u|∞)Ym

Next term in the e-equation is estimated by the product formula

(240) ‖∂m(fg)‖2 ≤ ‖f‖Hm |g|∞ + |f |∞‖g‖Hm .So, we have

|∂me∂m(e∇ · u)| ≤ ‖e‖2Hm|∇u|∞ + ‖e‖Hm |e|∞‖u‖Hm+1 ≤ (|e|∞ + |∇u|∞)Ym.

Finally,

|∂me[(∇ · u)2 − Tr(∇u)2]| ≤ ‖e‖Hm‖u‖Hm+1 |∇u|∞ ≤ |∇u|∞Ym.Thus,

(241)d

dt‖e‖2

Hm≤ (|e|∞ + |∇u|∞)Ym.

Next perform the main technical estimate on the velocity equation. We have

∂t‖u‖2Hm+1 = −∂m+1(u · ∇u) · ∂m+1u + ∂m+1Cφ(u, ρ) · ∂m+1u.

The transport term is estimated using the classical commutator estimate

∂m+1(u · ∇u) · ∂m+1u = u · ∇(∂m+1u) · ∂m+1u + [∂m+1,u]∇u · ∂m+1u

Then

u · ∇(∂m+1u) · ∂m+1u = −1

2(∇ · u)|∂m+1u|2 ≤ |∇u|∞‖u‖2Hm+1 ,

and using (217) we obtain

|[∂m+1,u]∇u · ∂m+1u| ≤ |∇u|∞‖u‖2Hm+1 .

Thus,

∂t‖u‖2Hm+1 ≤ |∇u|∞Ym + ∂m+1Cφ(u, ρ) · ∂m+1u.

Let us expand the commutator

∂m+1Cφ(u, ρ) =

m+1∑l=0

(m+ 1

l

)Cφ(∂lu, ∂m+1−lρ).

76 ROMAN SHVYDKOY

One end-point case, l = m+ 1, gives rise to a dissipative term:∫TnCφ(∂m+1u, ρ) · ∂m+1u dx = −1

2

∫T2n

φ(z)|δz∂m+1u(x)|2ρ(x+ z) dz dx

− 1

2

∫T2n

φ(z)δz∂m+1u(x)∂m+1u(x)δzρ(x) dz dx.

The first term is bounded by

−ρ−∫T2n

φ(z)|δz∂m+1u(x)|2 dz dx ∼ −ρ−‖u‖2Hm+1+α2,

which is the main dissipation term. The second is estimated as follows. Let us pick an ε > 0 so small that1 + α

2 > α+ ε. Then∣∣∣∣∫T2n

φ(z)δz∂m+1u(x)∂m+1u(x)δzρ(x) dz dx

∣∣∣∣ ≤ |∇ρ|∞ ∫T2n

|∂m+1δzu(x)||z|n/2+α−1+ε

|∂m+1u(x)||z|n/2−ε dz dx

≤ |∇ρ|∞‖u‖Hm+1‖u‖Hm+α+ε ≤ |∇ρ|∞‖u‖Hm+1‖u‖Hm+1+α/2 ≤ 1

2ρ−‖u‖2Hm+1+α/2 + ρ−1

− |∇ρ|2∞Ym,

where the first term is absorbed into dissipation. So,∫TnCφ(∂m+1u, ρ) · ∂m+1u dx . −ρ−‖u‖2Hm+1+α

2+ ρ−1− |∇ρ|2∞Ym.

Let us consider first the other end-point case of l = 0. In this case the density suffers a derivative overload.We apply the following “easing” technique:∫

TnCφ(u, ∂m+1ρ) · ∂m+1u dx =

∫T2n

φ(z)δzu(x)∂m+1ρ(x+ z)∂m+1u(x) dz dx.

Observe that

∂m+1ρ(x+ z) = ∂z∂mx ρ(x+ z) = ∂z(∂

mx ρ(x+ z)− ∂mx ρ(x)) = ∂zδz∂

mρ(x).

Let us now integrate by parts in z:∫TnCφ(u, ∂m+1ρ) · ∂m+1u dx =

∫T2n

∂zφ(z)δzu(x)δz∂mρ(x)∂m+1u(x) dz dx +

+

∫T2n

φ(z)∂u(x+ z)δz∂mρ(x)∂m+1u(x) dz dx := J1 + J2.

Let us start with the J2 first. By symmetrization,

J2 =

∫T2n

δz∂u(x)δz∂mρ(x)∂m+1u(x)φ(z) dz dx−

∫T2n

∂u(x)δz∂mρ(x)δz∂

m+1u(x)φ(z) dz dx

:= J2,1 + J2,2.

Term J2,1 will appear in a series of similar terms that we will estimate systematically below. The bound forJ2,2 is rather elementary:

J2,2 ≤ |∇u|∞‖u‖Hm+1+α/2 + ‖ρ‖Hm+α/2 ≤ ερ−‖u‖2Hm+1+α/2 + ρ−1− |∇u|2∞Ym.

Similar computation can be made for J1. Indeed, using that ∂zφ(z) is odd, by symmetrization, we have

J1 =1

2

∫T2n

∂zφ(z)δzu(x)δz∂mρ(x)δz∂

m+1u(x) dz dx.

Replacing |δzu(x)| ≤ |z||u|∞, the rest of the term is estimated exactly as J2,2.To summarize, we have obtained the bound∫

TnCφ(u, ∂m+1ρ) · ∂m+1u dx ≤ ερ−‖u‖2Hm+1+α/2 + ρ−1

− |∇u|2∞Ym.


Let us now examine the rest of the commutators Cφ(∂lu, ∂m+1−lρ) for l = 1, . . . ,m. After symmetrizationwe obtain∫

TnCφ(∂lu, ∂m+1−lρ) · ∂m+1u dx =

1

2

∫T2n

δz∂lu(x)δz∂

m+1−lρ(x)∂m+1u(x)φ(z) dz dx+

+1

2

∫T2n

δz∂lu(x)∂m+1−lρ(x)δz∂

m+1u(x)φ(z) dz dx := J1 + J2.

Estimates on the new terms, J1, J2 are a little more sophisticated as we seek to optimize distribution ofLp-norms inside their components. Notice that the case l = 1 corresponds to the previously appeared termJ2,1.

So, let us assume that l = 1, . . . ,m. We will distribute the parameters in J1 as follows

J1 =

∫T2n

δz∂lu(x)

|z|np+α2 +2δ

δz∂m+1−lρ(x)

|z|nq +α2

∂m+1u(x)

|z|n2−δ1

|z|nr−δ dz dx,

where δ > 0 is a small parameter to be determined later, and (2, p, q, r) is a Holder quadruple defined by

p = 2m+ α

2

l − 1 + α2

, q = 2m+ α− 1

m− l + α2

,1

r= 1− 1

2− 1

p− 1

q.

The existence of finite r is warranted by the strict inequality which is verified directly:

1

2+

1

p+

1

q< 1.

By the Holder inequality,

J1 ≤ ‖u‖W l+α2

+2δ,p‖ρ‖Wm+1−l+α2,q‖u‖Hm+1 .

Let us apply the following Gagliardo-Nirenberg inequalities to all the terms

‖u‖Hm+1 ≤ ‖u‖2m

2m+α

Hm+1+α2|∇u|

α2m+α

2 ≤ ‖u‖2m

2m+α

Hm+1+α2|∇u|

α2m+α∞

‖u‖W l+α

2+2δ,p ≤ ‖u‖θ1

Hm+1+α2|∇u|1−θ1∞

‖ρ‖Wm+1−l+α

2,q ≤ ‖ρ‖θ2Hm+α |∇ρ|1−θ2∞ ,

where

θ1 =l − 1 + α

2 − np + 2δ

m+ α2 − n

2

, θ2 =m− l + α

2 − nq

m+ α− 1− n2

.

The exponents satisfy the necessary requirements

1 ≥ θ1 ≥l − 1 + α

2 + 2δ

m+ α2

, 1 ≥ θ2 =m− l + α

2

m+ α− 1,

and in fact,

θ1 =l − 1 + α

2

m+ α2

+O(δ).

Now, we have

J1 ≤ ‖u‖2m

2m+α+θ1

Hm+1+α2‖ρ‖θ2Hm+α |∇u|

α2m+α+1−θ1∞ |∇ρ|1−θ2∞ .

By generalized Young,

J1 ≤ ερ−‖u‖2Hm+1+α/2 + ρ−1− ‖ρ‖θ2QHm+α(|∇u|

α2m+α+1−θ1∞ |∇ρ|1−θ2∞ )Q.

where Q is the conjugate to 2m2m+α + θ1. We have θ2Q < 2 as long as

(242) θ1 + θ2 < 2− 2m

2m+ α.

We in fact have even stronger inequality, θ1 + θ2 < 1 provided δ is small enough. So, we arrived at

J1 ≤ ερ−‖u‖2Hm+1+α/2 + ρ−1− pN (|∇ρ|∞, |∇u|∞)Ym,

for some polynomial pN .

78 ROMAN SHVYDKOY

Finally, moving on to J2, we distribute the exponents as follows

J2 ≤∫T2n

|δz∂lu(x)||z|np+2δ+α

2

|∂m+1−lρ(x)||z|nq−δ

|δz∂m+1u(x)||z|n2 +α

2

1

|z|nr−δ dz dx ≤ ‖u‖W l+δ+α

2,p‖ρ‖Wm+1−l,q‖u‖Hm+1+α

2.

Here we choose (r, p, q, δ) as follows

q = 2m+ α− 1

m− l , p = 2m+ α

2

l − 1 + α2

,1

r= 1− 1

2− 1

p− 1

q.

and δ is small. With these choices we proceed with the Gagliardo-Nirenberg inequalities

‖u‖W l+2δ+α

2,p ≤ ‖u‖θ1

Hm+1+α2|∇u|1−θ1∞

‖ρ‖Wm+1−l,q ≤ ‖ρ‖θ2Hm+α

|∇ρ|1−θ2∞ ,

where

θ1 =l − 1 + α

2 + 2δ

m+ α2 − n

2

=l − 1 + α

2

m+ α2

+O(δ), θ2 =m− l

m+ α− 1.

Now to achieve the bound

J2 ≤ ερ−‖u‖2Hm+1+α/2 + ρ−1− pN (|∇ρ|∞, |∇u|∞)Ym,

we have to make sure that θ1 + θ2 ≤ 1, which is true for small δ.We have proved the following a priori bound on u:

∂t‖u‖2Hm+1 ≤ −1

2ρ−‖u‖2Hm+1+α/2 + ρ−1

− pN (|∇ρ|∞, |∇u|∞)Ym.

Together with the previously established bounds we obtain

d

dtYm ≤ −

1

2ρ−‖u‖2Hm+1+α/2 + ρ−1

− pN (|∇ρ|∞, |∇u|∞, |e|∞)Ym.

This of course implies a Riccati inequality, provided m > n2 + 1:

d

dtYm ≤ CY Nm ,

and provides a priori bound independent of the viscosity coefficient. Thus, we can extend it to an intervalindependent of ε as well. By the compactness argument similar to the smooth kernel case, we obtain alocal solution in the same class as initial data and u ∈ L2Hm+1+α/2. In addition, we obtain a continuationcriterion – as long as |∇ρ|∞, |∇u|∞, |e|∞ remain bounded on [0, T0) the solutions can be extended beyondT0. However everything is reduced to a control over the first two quantities, because |e|∞ remains boundedas long as |∇u|∞ is in view of (239).

It is clear from the proof that (231) can be replaced with an integrability condition with some high powerdepending on m,n, α.

8.3. Notes and References. Local existence and small initial data results for smooth models appearedin various contexts in works of Ha, et al [43, 44]. Our Theorem 8.1 with continuation criterion in terms ofdivergence (223) was proved in [63], although criteria of type (223) were known in the context of hyperbolicconservation laws, see Grassin [41], Poupaud [80] and references therein.

Equation (229) was found in many difference contexts, for example, as a 1D model of the Surface Quasi-Geostrophic equation [24], as a model for porous media with non-local pressure [10]. The blowup was provedby Castro and Cordoba in [19]. Remarkably, this blowup example can be extended to all 0 < α < 2 inthe context of Euler Alignment system as shown by Arnaiz and Castro [4]. Earlier Tan [95] demonstratedgrowth of |ρ|C1 as t→∞ for a similar density configuration with 1 point vacuum.

Although in 1D local existence for singular models appeared in the first papers by Shvydkoy, Tadmor [90]for α ≥ 1, and Do et al [39] for 0 < α < 1, it was not until now when the multidimensional case was addressedwith proper continuation criterion, see upcoming Lear, Shvydkoy [64]. For singular topological models localexistence was presented in Reynolds, Shvydkoy [83] and it is much more technically involved due to activedependencies on the density in the kernel. No practical continuation criterion has been established in thiscase yet.

The issues of sharp threshold condition for multi-D smooth models and global well-posedness for non-vacuous periodic solutions for singular models remain outstanding and important open problems. The models


have not been studied numerically extensively. However, Mao, Li, Karniadakis [67] provided evidence thatat least for α = 0.5 in 1D and α = 1.2 in 2D the singular models produce very close match with the discretedynamics. Those particular values were gathered through machine learning technology and the use of dataclosely related to the StarFlag project [22, 21, 20]. Numerical simulations were performed on a time intervallong enough to observe flocking states.

80 ROMAN SHVYDKOY

9. One-dimensional theory

In dimension 1, most complete regularity theory of alignment models is available. In this chapter wediscuss some of its highlights.

In 1D the system is given by

(243)

ρt + (ρu)x = 0,

ut + uux =

∫Ω

φ(x− y)(u(y)− u(x))ρ(y) dy(x, t) ∈ Ω× R+.

Here Ω is either R or periodic circle T. The underlying theme in this chapter will be to relate regularity andflocking behavior of the system to the new conserved quantity which is available in dimension 1 and a fewother exceptional multi-D cases:

(244) e = ux + Lφρ,where Lφ takes form of either of the integral representations (171) or (172) depending on the context. Thisquantity, whose physical role will be illuminated later in Section 9.2 is a key ingredient in the 1D regularitytheory.

9.1. Smooth kernels: critical thresholds and stability. The system (243) possesses an extra conser-vation law provided φ is a convolution kernel, φ = φ(x− y). To see this we note that the alignment term inthe velocity equation is given by the commutator

Cφ(u, ρ) = φ ∗ (uρ)− uφ ∗ ρ.Using this commutator structure and by elementary manipulation with the equations we obtain

(245) et + (ue)x = 0, e = ux + φ ∗ ρ.Note that the Burgers’ transport tends to create shocks, while the alignment force creates a smoothingcounter balance mechanism. One would expect then that a threshold condition would separate singularbehavior from regular. For pure Burgers equation such condition is provided by positive initial slope ux ≥ 0.In view of the regularizing additional effect of the alignment such condition can be relaxed to e0 ≥ 0. To seethat let us rewrite the e-equation as a non-autonomous logistic ODE along characteristics:

(246) e = e(φ ∗ ρ− e)It is clear that the sign of e will be preserved pointwise. So , if e0(x0) < 0 at some point x0, then e < −e2,and consequently the solution blows up in finite time. On the other hand, supposing e0 ≥ 0 we have e(t) ≥ 0for all times, and e remains a priori bounded since φ∗ρ ≤ CM . This in particular implies that ux ∈ L∞t,x. Inview of Theorem 8.1, this ensures global existence of solutions. Moreover, we can also obtain a hydrodynamicversion of strong flocking. Let us make this precise.

Theorem 9.1 (Threshold for global existence). Consider the system (243) on R or T with smooth kernel.For any initial condition (u0, ρ0) ∈ Hm × (L1

+ ∩W 1,∞), m > n2 + 2, which satisfies the threshold condition

e0 ≥ 0 there exists a unique global solution (u, ρ) ∈ Cw([0,∞);Hm × (L1+ ∩W 1,∞). If e0(x) < 0 at some

point, then the solution blows up in finite time.Furthermore, suppose that φ has a heavy tail. Let the initial flock has compact support Supp ρ0 and e0 ≥ 0.

Then the solution flocks strongly in the following sense: there exists C, δ > 0 depending on φ, and initialdata, such that the velocity satisfies

(247) supx∈Supp ρ(t)

|u(x, t)− u|+ |ux(x, t)|+ |uxx(x, t)| ≤ Ce−δt.

and the density ρ converges to a traveling wave ρ in the metric of Cγ for any 0 < γ < 1:

(248) ‖ρ(·, t)− ρ(· − ut)‖Cγ ≤ Ce−δt, t > 0.

Proof. The global existence has already been proved in the preceeding remarks.Let us now assume that we have a global solution with e0 ≥ 0 and φ has a heavy tail. We know from

Theorem 7.1 that the diameter of the flock will remain finite, D. Then we can estimate the convolution frombelow: for any x ∈ Supp ρ(t):

φ ∗ ρ(t, x) ≥ φ(D)M := c0.


Solving the logistic ODI: e ≥ e(c0−e) we find that starting from some time t0 for all t > t0 and x ∈ Supp ρ(t)we have e(x, t) ≥ c0/2. With this in mind let us write the equation for ux as follows

(249)D

Dtux =

∫Rφ′(x− y)(u(y)− u(x))ρ(y) dy − e(x)ux(x).

We already know from Theorem 7.1 that the velocity fluctuations are decaying with exponential rate. Hence,the integral above will be bounded by |φ′|∞ME(t), where we denote by E any quantity that shows expo-nential decay. Thus, multiplying (249) with ux and evaluating at the maximum over Supp ρ0 we obtain

d

dt‖ux‖L∞(Supp ρ(t)) = E(t)− 1

2c0‖ux‖L∞(Supp ρ(t)).

This implies the desired result by integration. In view of (221) the density enjoys a pointwise global bound

(250) supt>0‖ρ(·, t)‖∞ = R <∞.

For the second derivative, we have

D

Dtuxx + 2uxuxx =

∫Rφ′′(x− y)(u(y)− u(x))ρ(y) dy − 2uxφ

′ ∗ ρ− euxx.(251)

We see that the integral term as well as ux is of type E(t) in view of the previously established bounds. Notealso that |φx ∗ ρ| ≤ |φx|∞M . So, we obtain

(252)d

dt‖uxx‖L∞(Supp ρ(t)) = E(t)− 1

2c0‖uxx‖L∞(Supp ρ(t)).

This implies exponential decay. Moving to the density, we have

(253) ∂tρx + uρxx = −2uxρx − uxxρ = Eρx + E.

This shows that ρx remains uniformly bounded. Now, to establish strong flocking we pass to the movingframe x− ut and write the continuity equation in new coordinates

(254)D

Dtρ = −(u− u)ρx − uxρ = E.

This shows that ρ(t) is Cauchy in t in the metric of L∞. Hence, there exists ρ ∈ L∞ such that ‖ρ(t)− ρ‖∞ =E(t). Since ρ′ is uniformly bounded, this also shows that ρ is Lipschitz. Convergence in Cγ , γ < 1, followsby interpolation.

Using stability estimate (189) we can easily conclude stability of the limiting flock distributions in thesense of KR-distance. Indeed, if ρ → ρ in Cγ then certainly W1(ρ, ρ) → 0. So, if we start from two flockswith the same mass and momentum and bounded supports, then using (189) we obtain (by translationinvariance of the KR-metric)

(255) W1(ρ′, ρ′′) ≤ C[W1(ρ′0, ρ′′0) + ‖u′0 − u′′0‖L∞(Ω)],

where Ω = Supp ρ′0 ∪ Supp ρ′′0 . This establishes a direct stability control of limiting flocks with respect toinitial perturbation. To bootstrap stability estimate to higher regularity class, let us note that in the courseof proof of Theorem 9.1 we established global bound on |ρx|∞ by a constant depending on initial data.Thus, ρ will remain in W 1,∞ with similar bound. Interpolating between W−1,∞ (which is equivalent toKR-distance, see Remark 7.2) and W 1,∞ gives a bound in Holder class Cγ , γ < 1:

(256) ‖ρ′ − ρ′′‖Cγ . [W1(ρ′0, ρ′′0) + ‖u′0 − u′′0‖L∞(Ω)]

1−γ2 .

9.2. Entropy. Csiszar-Kullback inequality. Distribution of the limiting flock. Although in thecase of symmetric communication the limiting velocity is determined from the initial condition by use ofmomentum conservation, the limiting shape of the density profile ρ is an emergent quantity which is notknown a priori. Yet it is easy to notice that the e-quantity is somehow involved in determining ρ. Let usassume in this section that e = ux + Lφρ, where Lφ is defined by (172) in either smooth or singular case.Let us write the continuity equation with the use of e:

(257) ρt + uρx + eρ = ρLφρ.

82 ROMAN SHVYDKOY

Let us assume for a moment that φ is a smooth absolute kernel on T. Suppose that e0 = 0, and hencee(t) = 0 for all time. Note that in this case ux + φ ∗ ρ ≥ ρ(x)|φ|1 ≥ 0, so the global solution exists. Then

ρt + uρx = ρLφρ,and so, ρ obeys the maximum principle. Let us assume that there is no vacuum ρ−(0) > 0. Let us thenwrite the equation for the new quantity ln ρ:

(ln ρ)t + u(ln ρ)x = Lφρ.Evaluating at a point of minimum we obtain

(ln ρ−)t =

∫φ(x, y)(ρ(y)− ρ−) dy ≥ c0(M − 2πρ−),

and at the maximum

(ln ρ+)t =

∫φ(x, y)(ρ(y)− ρ+) dy ≤ c0(M − 2πρ+).

Subtracting the two we obtain

d

dtlnρ+

ρ−≤ −2πc0(ρ+ − ρ−) ≤ −2πc0ρ−(0)

(ρ+

ρ−− 1

)≤ −2πc0ρ−(0) ln

ρ+

ρ−.

We conclude that

lnρ+

ρ−≤ c1e−c2t.

Since the maximum also stays bounded we have inequality

lnρ+

ρ−≥ c

(ρ+

ρ−− 1

),

So, we conclude that the density flattens out exponentially fast to a uniformly distributed state ρ = 12πM .

In view of this computation we can see that e is directly responsible for the flattening of the density. Itturns out that, first, a similar result is true for local kernels and even vacuous solutions. And second, ingeneral the size of e per mass, i.e. the quotient q = e

ρ , measures how far ρ is from the uniform distribution.

Thus, e plays the role of a topological entropy of the flock – a measure of disorder. We will address thisinterpretation in the next theorem.

Theorem 9.2. Let (ρ, u) be a smooth solution to the system (243) on the 1D torus T, and φ is a smoothlocal kernel:

φ(r) ≥ λ1r<r0 .If e0 = 0, then

(258) ‖ρ(t)− ρ‖1 ≤ c1(‖ρ0‖2)e−c2(λ,r0,M,‖ρ0‖∞)t,

where ρ = 12πM .

In general, provided ‖q0‖∞ < ‖φ‖1, one has

(259) lim supt→∞

‖ρ(·, t)− ρ‖1 ≤M‖q0‖∞‖φ‖∞

λc(r0)(‖φ‖1 − ‖q0‖∞),

Let us note that the dependence on ‖q0‖∞ is linear for small values. At the same time, the bound isinversely proportional to the strength λ, which shows the stabilizing effect of communication on the structureof the flock. Let us note that q satisfies the transport equation

(260)d

dtq + uqx = 0, q =

e

ρ,

and hence the value of ‖q‖L∞ is preserved for all time.The proof will be split in the following few subsections. First, let us recall a very useful tool – The

Csiszar-Kullback inequality.


9.2.1. The Csiszar-Kullback inequality. Let us consider two functions f ≥ 0, g > 0 on a measure space(Ω,Σ, µ), such that ∫

Ω

f(x) dµ(x) =

∫Ω

g(x) dµ(x).

Then

(261)

∫Ω

|f − g|2 dµ

g≥∫

Ω

f logf

gdµ ≥ 1

8‖f − g‖2L1 .

Proof. Let us start from an elementary inequality:

x(x− 1) ≥ x log x ≥ (x− 1) +1

2(x− 1)21x<1.

The upper inequality is elementary:∫Ω

f logf

gdµ ≤

∫Ω

f

[f

g− 1

]dµ =

∫Ω

(f − g)

[f

g− 1

]dµ =

∫Ω

|f − g|2 dµ

g.

Let us prove the lower inequality. On the one end,∫Ω

f logf

gdµ(x) ≥

∫Ω

(f − g) dµ(x)︸︷︷︸=0

+1

2

∫f<g

g(f/g − 1)2 dµ(x),

and on the other end,

‖f − g‖L1 =

∫f<g

(g − f) dµ+

∫g≤f

(f − g) dµ =

∫f<g

(g − f) dµ−∫f<g

(f − g) dµ = 2

∫f<g

(g − f) dµ.

Considering g dµ as a probability measure, we use the Holder inequality:∫f<g

(g − f) dµ =

∫f<g

(1− f/g)g dµ ≤(∫

f<g

|1− f/g|2g dµ

) 12

.

Connecting the two ends produces lower inequality in (261).

The object to our study will by the relative entropy defined by

(262) H =

∫Tρ log

ρ

ρdx =

∫Tρ log ρdx−M log ρ,

By rescaling the Csiszar-Kullback inequality applied to f = ρ/M , g = 1/2π on T we obtain

(263)1

16π‖ρ− ρ‖2L1 ≤ ρH ≤ ‖ρ− ρ‖2L2 .

9.2.2. Evolution of the entropy. At the heart of the argument is the equation for the entropy (262) whichone obtains testing the continuity equation with log ρ+ 1:

(ρ log ρ)t = ρt(log ρ+ 1) = −(ρu)′(log ρ+ 1)

= −ρ′(log ρ+ 1)u− ρu′(log ρ+ 1)

= −(ρ log ρ)′u− (ρ log ρ)u′ − ρu′

= −[u(ρ log ρ)]′ − ρu′ = −[u(ρ log ρ)]′ − ρ2q + ρLψρ.Therefore,

(264)dHdt

=d

dt

∫Tρ log ρ dx = −

∫Tρ2q dx−

∫T2

φ(x− y)(ρ(x)− ρ(y))ρ(x) dxdy.

Noting that∫T ρq dx =

∫T edx = 0, we can subtract ρ from one density in the first integral on the left hand

side. After additionally symmetrizing the last integral we obtain

(265)dHdt

= −∫T

(ρ− ρ) ρq dx− 1

2

∫T2

φ(x− y)|ρ(x)− ρ(y)|2 dx dy.

84 ROMAN SHVYDKOY

9.2.3. Bounds on the dissipation. If our kernel was absolute, it would be easy to get a positive lower boundon the dissipation term:

(266)

∫T2

φ(x− y)|ρ(x)− ρ(y)|2 dxdy ≥ (inf φ)

∫T2

|ρ(x)− ρ(y)|2 dxdy = 2(inf φ)‖ρ− ρ‖22.

Since we have a non-trivial lower bound on the kernel only near the diagonal (x, y) ∈ T2 : |x− y| < r0, weneed a substitute for (266) stated in the following Lemma.

Lemma 9.3. The following inequality holds:

(267)

∫∫|x−y|<r0

|ρ(x)− ρ(y)|2 dy dx ≥ c(r0)‖ρ− ρ‖22.

Proof. Denote by χ be any non-negative bump function supported on Br0(0), constant on Br0/2 and∫χ(r) dr = 1. Then on the Fourier side, χ(0) = 1 and |χ(k)| < 1 for all k ∈ Z\0. On the other

hand, by the Riemann-Lebesgue Lemma, χ(k) → 0 as k → ∞. Therefore, |χ(k)| ≤ 1 − ε for some ε > 0depending only on r0 (k 6= 0). Define ρr0(x) = χ ∗ ρ(x), so that

(ρ− ρr0 ) (k) = (1− χ(k))ρ(k).

Hence,|(ρ− ρr0 ) (k)| ≥ ε|ρ(k)|, k ∈ Z, k 6= 0,

and ρ(0) = ρr0(0). Consequently,

‖ρ− ρ‖22 =∑

k∈Z\0|ρ(k)|2 ≤ ε−2

∑k∈Z|(ρ− ρR0

) (k)|2 = ε−2‖ρ− ρR0‖2L2 .

By∫T χ = 1 and the Minkowski inequality,

‖ρ− ρr0‖2L2 =

∥∥∥∥∫Tχ(y)(ρ(·)− ρ(· − y)) dy

∥∥∥∥2

2

≤∫|y|<r0

‖ρ(·)− ρ(· − y)‖22 dy

=

∫T

∫|z|<r0

|ρ(x)− ρ(x+ z)|2 dz dx.

Combining the above we obtain

‖ρ− ρ‖22 ≤ ε−2

∫T

∫|z|<r0

|ρ(x)− ρ(x+ z)|2 dz dx.

Choosing c(r0) = ε2 concludes the proof.

By virtue of the lemma, the dissipation term has the following lower bound

(268)1

2

∫T2

φ(x− y)|ρ(x)− ρ(y)|2 dxdy ≥ c‖ρ− ρ‖22.

9.2.4. The entropy equation revisited. We now return to the entropy equation (265). We have

H(t) ≤ ‖ρ(t)‖L∞‖q0‖L∞‖ρ(·, t)− ρ‖L1 − c‖ρ(·, t)− ρ‖2L2

≤ ‖ρ(t)‖L∞‖q0‖L∞√

16πρH(t)− cρH(t).

Setting Y =√H, we obtain

Y (t) ≤ ‖ρ(t)‖L∞‖q0‖L∞√πρ− cρY (t).

By Gronwall’s lemma we arrive at

(269) Y (t) ≤ Y0e−cρt +

√πρ‖q0‖L∞

∫ t

0

‖ρ(s)‖L∞e−cρ(t−s) ds.

It is now easy to reach the conclusion of Theorem 9.2. If e0 ≡ 0, then the second term in (269) drops outcompletely and (263) closes this particular case.

For general e, we have

(270) lim supt→∞

‖ρ(·, t)− ρ‖L1 ≤M‖q0‖L∞ lim supt→∞

∫ t

0

‖ρ(s)‖L∞e−cρ(t−s) ds ≤ ‖q0‖L∞λc(r0)

lim supt→∞

‖ρ(t)‖L∞ .


Thus the proof of Theorem 9.2 is reduced to estimating the density amplitude.

9.2.5. Bounds on the Density Amplitude. First let us observe that if X(t) ≤ AX(t)[B −X(t)], where A andB are positive constants and X(t) is a positive function, then

(271) X(t) ≤ BX(0)

X(0) + (B −X(0)) exp(−ABt) .

In particular, lim supt→∞X(t) ≤ B.Let ρ+(t) denote the maximum ρ at time t, and x+ be a point where the maximum is achieved. Then if

‖q0‖L∞ < ‖φ‖L1 , one can get an upper bound on ‖ρ(t)‖L∞ by integrating the differential inequality derivedbelow.

d

dtρ+(t) = −ρ+(t)u′(x+, t) = −ρ+(t)2q(x+, t) + ρ+(t)

∫Tφ(x+ − y)(ρ(y, t)− ρ+(t)) dy

≤ (‖q0‖∞ − ‖φ‖1)ρ+(t)2 +M‖φ‖∞ρ+(t)

= (‖φ‖1 − ‖q0‖∞)ρ+(t)

[M‖φ‖∞

‖φ‖1 − ‖q0‖∞− ρ+(t)

].

In view of (271) we obtain

lim supt→∞

‖ρ(t)‖∞ ≤M‖φ‖∞

‖φ‖1 − ‖q0‖∞.

Plugging into (270) we conclude

lim supt→∞

‖ρ(·, t)− ρ‖L1 ≤ M‖q0‖∞‖φ‖∞λc(r0)(‖φ‖1 − ‖q0‖∞)

.

9.3. Alignment on T with degenerate kernel. The corrector method. On the periodic circle, thereis a mechanism for alignment with local communication even for vacuous solutions. It is clear already onthe level of particle dynamics – if two agents have not yet aligned and are past their communication range,they would meet at the opposite end of the circle and reestablish communication. The alignment can beestablished in this case with an adaptation of the corrector method we introduced in Section 3.7.

Theorem 9.4. For any solution of either the discrete or hydrodynamic system on T the following holds

(i) For sub-quadratic communication

(272) λ1r<r0 ≤ φ(r) ≤ Λ

r2,

one has

(273) V2(t) ≤ C ln t

t,

as t→∞, where C depends only on the initial condition.(ii) If the kernel satisfies the more singular assumption

(274) 1r<r0λ

rβ≤ φ(r) ≤ Λ

rβ, β > 2

then we can only conclude V2(t)→ 0, t→∞.

Let us note that this result does not improve the rate in the topological models since the variations of thedensity preclude us from making assumption (272).

Proof. The proof will be carried out in the discrete case only since the continuous version is entirely similar.We will limit ourselves to providing necessary modifications in that case.

We go back to the basic energy law (42) and construct a corrector functional G which serves to compensatefor the missing interactions. To do that we first define a periodic analogue of the directed distance:

dij(t) = −xij sgn(vij) mod 2π,

86 ROMAN SHVYDKOY

bb b b

−r0 r0 2π − r0−2π + r0

ψ(r)

Figure 12. Slope kernel

where xi, xj ∈ [0, 2π) are viewed on the same coordinate chart. The distance picks up the length of thearch between xi and xj which contracts under the evolution of the agents. The distance undergoes jumpdiscontinuities at xi = xj and vi = vj . At any other point, we have

(275)d

dtdij = −|vij |.

Next, we define a slope kernel ψ ≥ 0 as follows (see Figure 12)

ψ(x) =

−x+ r0, − r0 ≤ x ≤ r0

r0

π − r0x− r2

0

π − r0, r0 < x < 2π − r0,

extended periodically on R. Finally, we define the corrector

G(t) =1

N2

N∑i,j=1

|vij |ψ(dij).

Let us look into differentiability of G:

d

dtG = − 1

N2

N∑i,j=1

|vij |2ψ′(dij) +2

N3

N∑i,j=1

ψ(dij)sgn(vij)

N∑k=1

vkiφki.

The formula can be justified classically, at those times when there is no jump, i.e. xi 6= xj and vi 6= vj , dueto (275). When two agents pass each other xi = xj we use periodicity of ψ, and when vij = 0, the factor|vij | vanishes.

We continue

d

dtG(t) =

1

N2

N∑i,j=1

|vij |2 1|xij |≤r0 −r0

π − r0

1

N2

N∑i,j=1

|vij |2 1|xij |≥r0 +R,

where

(276) R =2

N3

N∑i,j,k=1

ψ(dij)sgn(vij)vkiφki.

Symmetrizing over i, k, we obtain

R =1

N3

N∑i,j,k=1

(ψ(dij)sgn(vij)− ψ(dkj)sgn(vkj))vkiφki.


In the case vi ≥ vj ≥ vk or vi ≤ vj ≤ vk, the summand is negative, and so we can neglect it. Continuing,

R ≤ 1

N3

N∑i,j,k=1

(ψ(dij)sgn(vij)− ψ(dkj)sgn(vkj)

)vkiφki 1vj>max(vi,vk)

+1

N3

N∑i,j,k=1

(ψ(dij)sgn(vij)− ψ(dkj)sgn(vkj)

)vkiφki 1vj<min(vi,vk)

=1

N3

N∑i,j,k=1

(ψ(dkj)− ψ(dij)

)vkiφki 1vj>max(vi,vk)

+1

N3

N∑i,j,k=1

(ψ(dij)− ψ(dkj)

)vkiφki 1vj<min(vi,vk).

In the cases vj > max(vi, vk) and vj < min(vi, vk), we see that vi − vj and vk − vj have the same sign sodij and dkj are computed in the same direction. So, by the Lipschitz continuity of ψ and by the triangleinequality we find that |ψ(dij)− ψ(dkj)| ≤ C|xi − xk|. Therefore,

R ≤ C

N3

N∑i,j,k=1

|xik|vikφki =C

N2

N∑i,k=1

|xik|vikφki ≤1

t

C

bN2

N∑i,k=1

|xik|2φki + tb

N2

N∑i,k=1

v2ikφki.

Let us proceed now under the assumption of (i). Here we obtain

R ≤ c

t+ btI2.

Then the corrector equation becomes

d

dtG(t) ≤ aI2 + b(tI2 − V2) +

c

t.

Let us form form another functional:L = G + btV2 + aV2.

It satisfies inequality ddtL ≤ c

t . Thus, L(t) . ln t, and the resulting bound follows.

For part (ii) we use the collision potential (38) with precomputed bound in (39),

R ≤ C

1

N2

N∑i,k=1

|xik|2−β |xik|βφki

1/2√I2 .

√I2

√C ≤ c1

√I2(t) + c2

√I2(t)

∫ t

0

√I2(s) ds.

We can replace by the generalized Young inequality,

c1√I2(t) ≤ c3

t+ btI2(t),

and obtaind

dtG(t) ≤ aI2 + b(tI2 − V2) +

c3t

+ c2√I2(t)

∫ t

0

√I2(s) ds.

With L being defined as before, we continue

d

dtL . c3

t+ c2

√I2(t)

∫ t

0

√I2(s) ds.

Integrating over [0, T ],

L(T ) . L(0) + lnT +

(∫ T

0

√I2(s) ds

)2

.

Thus,

V2(T ) ≤ 1

TL(T ) .

lnT

T+

1

T

(∫ T

0

√I2(s) ds

)2

.

The right hand side tends to zero which can be readily seen by splitting the integral into (0, T ′) and (T ′, T ),where T ′ is large.

88 ROMAN SHVYDKOY

We already noted that the hydrodynamic version of the result is identical. Let us make some remarksabout the proof. We work with the Lagrangian formulation (177). As in the discrete case we define thedirected distance

dαβ(t) = (x(α, t)− x(β, t)) sgn(v(β, t)− v(α, t)) mod 2π,

and the corrector with ψ as before:

G =

∫T2

|uαβ |ψ(dαβ) dm0(α, β).

We calculate the derivative of G:

d

dtG = −

∫T2

|uαβ |2ψ′(dαβ) dm0(α, β) +

∫T3

sgn(uαβ)ψ(dαβ)φαγuγα dm0(α, β, γ)

≤ aI2 − bV2 +R.Here,

R =

∫T3

sgn(uαβ)ψ(dαβ)φαγuγα dm0(α, β, γ)

=1

2

∫T3

[sgn(uαβ)ψ(dαβ)− sgn(uγβ)ψ(dγβ)]φαγuγα dm0(α, β, γ)

≤∫T3

(ψ(dαβ)− ψ(dγβ))φαγuγα1u(β)<minu(α),u(γ) dm0(α, β, γ)

+

∫T3

(ψ(dγβ)− ψ(dαβ))φαγuγα1u(β)>maxu(α),u(γ) dm0(α, β, γ)

≤∫T2

|xαγ ||uγα|φαγ dm0(α, γ).

In case (i) we obtain

R ≤ c

t+ btI2,

and the proof concludes as in the agent-based settings. In case (ii) we consider the collisional potential

C =

∫T2

dm0(α, β)

(|xαβ | ∧ r0)β−2.

It is well-posed for β < 3 (in view of also the fact that the density is bounded for regular solutions). Asimilar computation establishes (39), and from this point on the proof is ad verbatim.

In hydrodynamic settings the L2-based alignment result does not provide sufficient information for point-wise behavior. So, it is desirable to obtain L∞-based alignment statement in this context. The mechanismfor such alignment comes from considering regions where the density is non-negligible – here the alignmentterm works faster than the transport to avoid agent collisions. At the same time if density is thin the equa-tion acts as classical Burgers’ equation. So, in order to avoid a blowup it must have low velocity fluctuations.In other words, it has to be aligned sufficiently well.

Theorem 9.5. Consider the system (243) on T with a smooth non-trivial non-negative kernel. Then anyglobal classical solution aligns:

(277) supx|u(x, t)− u| ≤ C

(ln t

t

) 15

.

Proof. By the Galilean invariance we can assume throughout that u = 0. As a consequence of the energyequality and (273) we obtain∫ ∞

T

∫T2

φαβ |v(α, t)− v(β, t)|2 dm0(α, β) dt ≤ C lnT

T:= ε.

Here we passed to Lagrangian coordinates v(α, t) = u(x(α, t), t). Denote

I2(α, T ) =

∫ ∞T

∫Tφαβ |v(α, t)− v(β, t)|2 dm0(β) dt.


So, we have ∫TI2(α, T ) dm0(α) ≤ ε.

Let us fix another small parameter δ > 0 and define the “good set”:

Gδ(T ) = α : I2(α, T ) ≤ δ.We denote by Gcδ the complement of Gδ, so that m0(Gcδ) = M −m0(Gδ) (recall that M is the total mass ofthe flock). By the Chebychev inequality,

(278) m0(Gcδ) <ε

δ.

Thus, the good set occupies almost all of the domain provided ε δ. We now proceed by proving thatalignment occurs first on the good set identified above, and then on the rest of the torus later in time withina controlled time scale.

Lemma 9.6 (Alignment on Gδ). We have

supα1,α2∈Gδ(T ), t≥T

|v(α1, t)− v(α2, t)| . δ2/3.

Proof. It suffices to establish alignment at time T only because of monotonicity of the our I2-function:

I2(α, t) ≤ I2(α, T ), t > T,

which in particular implies that the good sets are increasing in time, Gδ(T ) ⊂ Gδ(t).Integrating the equation Euler-Alignment system

d

dtv(α, t) =

∫Tφαβvαβ dm0(β)

over [T, t] for any α ∈ Gδ we obtain

(279) |v(α, t)− v(α, T )| ≤∫ t

T

∫Tφαβ |vαβ |dm0(β) . δ

√t− T .

Assume that for some α1, α2 ∈ Gδ we have

v(α1, T )− v(α2, T ) > U,

where U to be determined later. Then in view of (279),

v(α1, t)− v(α2, t) >U

2,

so long as

(280) t− T . U2

δ2.

During this time interval the corresponding characteristics will undergo a significant displacement

x(α1, t)− x(α2, t) ≥ x(α1, T )− x(α2, T ) +1

2U(t− T ) mod 2π,

where 12U(t − T ) > 4π as long as t − T & 1

U . If this is allowed to happen, then the characteristics willfind themselves at the separation distance equal to 2π = 0 at some point in time, which means they wouldcollapse. We then obtain

(281)1

U&U2

δ2,

which gives U . δ2/3 as claimed.

On the next step we show that our solution aligns at a certain not too remote later time t > T .

Lemma 9.7 (Alignment outside Gδ). For all t & T + 1δ1/3+(ε/δ)1/2 we have

supα∈T,γ∈Gδ(T )

|v(α, t)− v(γ, t)| . δ1/3 + (ε/δ)1/2.

90 ROMAN SHVYDKOY

Proof. Fix α ∈ T and γ ∈ Gδ(T ). Let us write

d

dtv(α, t) =

∫Tvβαφαβ dm0(β) =

∫T(vβγ + vγα)φαβ dm0(β)

= (φ ∗ ρ)(x(α, t), t)vγα +

∫Tvβγφαβ dm0(β).

The integral term on the right hand side above will remain small for all t ≥ T , by virtue of Lemma 9.6 and(278). Indeed,∣∣∣∣∫

Tvβγ(t)φαβ dm0(β)

∣∣∣∣ =

∣∣∣∣∣∫Gδ(T )

vβγ(t)φαβ dm0(β)

∣∣∣∣∣+

∣∣∣∣∣∫Gcδ(T )

vβγ(t)φαβ dm0(β)

∣∣∣∣∣. δ2/3 +

ε

δ.

Thus,

(282) (φ ∗ ρ)vγα − δ2/3 − ε

δ≤ d

dtv(α, t) ≤ (φ ∗ ρ)vγα + δ2/3 +

ε

δ.

Let us consider a fixed time t & T + 1δ1/3+(ε/δ)1/2 , and assume that vαγ(t) = U > 0 for some U to be

determined later. Let us now reverse the dynamics backwards in time from the moment t. For a time period[s, t], where T < s < t, the difference will remain positive vαγ(s) > 0. On that time period, the right handside of (282) implies

d

dtv ≤ δ2/3 +

ε

δ

and hence,

v(α, t)−(δ2/3 +

ε

δ

)(t− s) ≤ v(α, s).

Simultaneously, by (279) applied for γ ∈ Gδ, we obtain

|v(γ, t)− v(γ, s)| ≤ δ(t− s)1/2.

In combination with the previous, this implies

U −(δ2/3 +

ε

δ

)(t− s)− δ(t− s)1/2 = vαγ(t)−

(δ2/3 +

ε

δ

)(t− s)− δ(t− s)1/2 ≤ vαγ(s).

We find that

vαγ(s) ≥ U

2,

as long as (t − s) . Uδ2/3+ ε

δ

and (t − s) . U2

δ2 . The former condition is more restrictive, unless U . δ4/3,

in which case we have reached our goal. Arguing as in Lemma 9.6 we obtain collision backwards in time,provided (t − s) ∼ 1/U . This becomes possible if U & δ1/3 + (ε/δ)1/2 on the time interval of lengtht− T & 1/U , which is true under the assumption.

Arguing similarly from the opposite end, vαγ(t) = −U < 0, we obtain the bound from below.

Lemma 9.7 implies the following quantified global alignment starting from t & T + 1δ1/3+(ε/δ)1/2

supα,γ∈T

|v(α, t)− v(γ, t)| . δ1/3 + (ε/δ)1/2.

Optimization over δ, produces the choice δ = ε3/5. Recalling that ε = lnT/T , we obtain

supα,γ∈T

|v(α, t)− v(γ, t)| .(

lnT

T

)1/5

,

for t ∼ T +(T

lnT

)1/5 ∼ T . This concludes the proof.


9.4. Singular models: global wellposedness. In this section we establish global well-posedness of solu-tions to the Euler alignment system (243) on the torus T for the case of singular kernel φ. More specifically,we assume that φ is the kernel of the classical fractional Laplacian Λα:

φ(z) =∑k∈Z

1

|z + 2πk|1+α, 0 < α < 2.

Very often we refer to φ as the kernel over the line φ(z) = 1|z|1+α which is justified by extending corresponding

functions periodically to the whole line:

Λαf(x) = p.v.

∫Tφ(z)δzf(x) dz = p.v.

∫Rδzf(x)

dz

|z|1+α.

The analysis can be carried out for local metric kernels as well in a similar fashion.As always in 1D the corresponding entropy will play a key role in establishing regularity of solutions to

(243):

(283) e = ux + Λαρ.

The main result is the following.

Theorem 9.8. Suppose m ≥ 3 and 0 < α < 2. Let (u0, ρ0) ∈ Hm+1(T)×Hm+α(T), and ρ0(x) > 0 for allx ∈ Tn. Then there exists a unique non-vacuous global in time solution to (243) in the class

(284) u ∈ Cw([0,∞);Hm+1) ∩ L2([0,∞); Hm+1+α/2), ρ ∈ Cw([0,∞);Hm+α).

Moreover, the solution obeys uniform bounds on the density

(285) c0 ≤ ρ(x, t) ≤ C0, t ≥ 0,

and strong flocking: ρ ∈ Hm+α such that

(286) ‖u(t)− u‖W 2,∞ + ‖ρ(·, t)− ρ(· − ut)‖Cγ ≤ Ce−δt t > 0, 0 < γ < 1.

Proof. According to our local well-posedness Theorem 8.5 we already have a local solution (u, ρ) on timeinterval [0, T0). We proceed in several steps. First, we establish uniform bounds (285) on the density whichdepend only on the initial conditions. So, such bounds hold uniformly on the available time interval [0, T0).Next, we invoke results from the theory fractional parabolic equations to conclude that our solution gainsHolder regularity after a short period of time, and the Holder exponent as well as the bound on the Holdernorm depend on the L∞ bound of the solution. Finally, we establish a continuation criterion much weakerthan that of Theorem 8.5 – claiming that any Holder regularity of the density propels higher order normsbeyond T0. Here the case α = turns out to be more challenging than the rest of the range.

Paired with the mass equation we find that the ratio q = e/ρ satisfies the transport equation

(287)D

Dtq = qt + uqx = 0.

Starting from sufficiently smooth initial condition with ρ0 away from vacuum we can assume that

(288) |q(t)|∞ = |q0|∞ <∞.

Step 1: bounds on the density. We start by establishing (285) on the given time interval.First, recall that q = e

ρ is transported, see (260), and hence is bounded for all time with its initial value

|q0|∞. So, we can write the continuity equation as

(289) ρt + uρx = −qρ2 + ρΛα(ρ).

Let us evaluate at a point x+ where the maximum of ρ, denoted ρ+, is reached. We obtain

d

dtρ+ = −q(x+, t)ρ

2+ + ρ+

∫φ(|z|)(ρ(x+ + z, t)− ρ+) dz

≤ |q0|∞ρ2+ + ρ+

∫|z|<r

φ(|z|)(ρ(x+ + z, t)− ρ+) dz

≤ |q0|∞ρ2+ +

1

r1+αρ+(M − 2rρ+) = |q0|∞ρ2

+ +1

r1+αMρ+ −

2

rαρ2

+.

92 ROMAN SHVYDKOY

Let us pick r small enough so that 2rα > |q0|∞ + 1. Then

(290)d

dtρ+ ≤ −ρ2

+ + C(M, r)ρ+,

which establishes the upper bound by integration.As to the lower bound we argue similarly. Let ρ− be the minimum value of ρ and x− a point where such

value is achieved. We have

d

dtρ− ≥ −|q0|∞ρ2

− + ρ−

∫Tφ(|z|)(ρ(x− + z, t)− ρ−) dz

≥ −|q0|∞ρ2− + φ−ρ−(M − 2πρ−) = −c1ρ2

− + c2ρ−.(291)

This readily implies the bound from below. Note that at this point the global communication of the modelis crucial: φ− > 0.

As a consequence of the lower bound on the density we have a global bound on the entropy:

(292) supt∈[0,T0)

|e(t)|∞ <∞.

Step 2: Holder regularization. The representation of continuity equation in the form (289) puts itinto the class of forced fractional parabolic equations with bounded drift and force:

∂tv + u · ∇v = L[v] + f,

where L has kernel

K(x, z, t) = ρ(x)1

|z|1+α,

which is even with respect to z. The bounds on the density provide uniform ellipticity bounds on the kernel1

|z|1+α . K(x, z, t) . 1|z|1+α .

With these ingredients at hand, the case α = 1 falls under the assumptions of Silverstre’s results [93]which provides Holder regularization bound given by

(293) |ρ|Cγ(T×[T0/2,T0)) ≤ C(|ρ|L∞(T×[0,T0)) + |ρe|L∞(T×[0,T0))),

for some γ > 0.The case α < 1 falls under the same result provided u ∈ L∞([0, T0);C1−α). This is indeed the case as

follows fromΛ−1α ∂xu = Λ−1

α e− ρ ∈ L∞t,x.Note that Λ−1

α ∂x a (1− α)-order differential operator.Finally, for α > 1 the Holder continuity follows from a similar identity for ρ:

Λα−1ρ = Λ−11 e−Hu,

where H is the Hilbert transform. Note that it sends functions in L∞ to B0∞,∞. Hence, ρ ∈ Bα−1

∞,∞ = Cα−1.

Step 3: Continuation and flocking. Last step is to show that if the density is bounded in Cγ ontime interval [T0/2, T0) then the solution remains uniformly in W 1,∞, and hence the continuation criterionof Theorem 8.5 applies. While doing that we will keep track of estimates on the W 1,∞ with the purpose toobtaining long time asymptotics.

Step 3a: control over ρ′. So, let us start with ρ′:

(294) ∂tρ′ + uρ′′ + u′ρ′ + e′ρ+ eρ′ = ρ′Λαρ+ ρΛαρ

′.

Using again u′ = e− Λαρ we rewrite

∂tρ′ + uρ′′ + e′ρ+ 2eρ′ = 2ρ′Λαρ+ ρΛαρ

′.

Evaluating at the maximum of |ρ′| and multiplying by ρ′ we obtain

(295) ∂t|ρ′|2 + e′ρρ′ + 2e|ρ′|2 = 2|ρ′|2Λαρ+ ρρ′Λαρ′.

Let us note that q′ satisfies the continuity equation, and consequently, q′

ρ is transported. So, |q′| ≤ Cρ

pointwise. For the e-quantity itself this implies pointwise bound

(296) |e′(x, t)| ≤ C(|ρ′(x, t)|+ ρ(x, t)).


Let us note that in order to make pointwise evaluation possible in (296) one has to assume regularitye′ ∈ C(T) which guaranteed provided m ≥ 2. With this at hand, and in view of (285) and (292) we canbound

|e′ρρ′ + 2e|ρ′|2| ≤ C(|ρ′|2 + |ρ′|).Thus,

(297) ∂t|ρ′|2 = C(|ρ′|2 + |ρ′|) + 2|ρ′|2Λαρ+ ρρ′Λαρ′.

Due to the bound from below on ρ, we estimate

(298) ρρ′Λαρ′ ≤ c1

∫R

(ρ′(x+ z)− ρ′(x))ρ′(x+ z)

|z|1+αdz ≤ −c2Dαρ

′(x).

where

Dαρ′(x) =

∫R

|ρ′(x)− ρ′(x+ z)|2|z|1+α

dz.

Lemma 9.9 (Non-local maximum principle). The following pointwise bound holds

(299) Dαρ′(x) ≥ c |ρ

′(x)|2+α

|ρ|α∞.

Proof. Fix an r > 0 to be determined later. We write

Dαρ′(x) ≥

∫|z|>r

|ρ′(x)− ρ′(x+ z)|2|z|1+α

dz ≥∫|z|>r

|ρ′(x)|2 − 2ρ′(x+ z)ρ′(x)

|z|1+αdz

=|ρ′(x)|2rα

− 2ρ′(x)

∫|z|>r

ρ′(x+ z)

|z|1+αdz.

Integrating by parts in the last integral we further estimate

Dαρ′(x) ≥ |ρ

′(x)|2rα

− cα|ρ′(x)||ρ|∞1

r1+α.

Choosing r = C |ρ|∞|ρ′(x)| , where C is large proves the esimate.

In view of density bounds we have a priori (285), the non-local maximum principle yields the followingnon-linear bound

Dαρ′(x) ≥ c|ρ′(x)|2+α.

We arrive at

(300) ∂t|ρ′|2 = C(|ρ′|2 + |ρ′|) + 2|ρ′|2Λαρ− c|ρ′|2+α − 1

2Dαρ

′(x).

The lower order terms |ρ′|2 + |ρ′| can be absorbed into dissipation by the generalized Young inequality:

|ρ′|2 + |ρ′| ≤ cε + ε|ρ′|2+α,

for ε > 0 small. So, it remains to obtain estimate on the remaining term |ρ′|2Λαρ.To do that we fix a scale parameter 1 > r > 0 to be determined later, and split the integral representation

of the fractional Laplacian into three parts: short-range, mid-range, and long-range

Λαρ(x) =

∫|z|<r

[δzρ(x)− ρ′(x)z]dz

|z|1+α+

∫r<|z|<1

δzρ(x)dz

|z|1+α+

∫|z|>1

δzρ(x)dz

|z|1+α

:= I + II + III.

For the short-range we use the dissipation directly:

(301) |ρ(x+ z)− ρ(x)− ρ′(x)z| =∣∣∣∣∫ z

0

(ρ′(x+ w)− ρ′(x)) dw

∣∣∣∣ ≤√Dαρ′(x)|z|1+α2 ,

so,

|I| ≤ r1−α/2√Dαρ′(x).

In the mid-range we use the available Holder continuity (here we can assume without loss of generality thatγ < α):

|II| ≤ |ρ|Cγrγ−α . rγ−α.

94 ROMAN SHVYDKOY

And finally, for the long-range we simply use the boundedness of ρ:

|III| . |ρ|∞.The competition occurs only between the short- and mid-range terms. Optimizing over r we set r =

(Dαρ′(x))−

12+α+2γ unless such expression is > 1, in which case we have an absolute bound on the dissipation

and the proof proceeds trivially. With the established bounds we obtain the following pointwise estimate

|Λαρ(x)| . c1 + c2(Dαρ′(x))

α−γ2+α+2γ .

Note that α−γ2+α+2γ <

α2+α . So, we can use the generalized Young inequality we obtain

|ρ′|2|Λαρ| . cε + ε|ρ′|2+α + εDαρ′(x).

Plugging this into (300) we arrive at

(302) ∂t|ρ′|2 ≤ c1 − c2|ρ′|2+α.

This concludes the proof of uniform bound ρ ∈ L∞([0, T0);W 1,∞).

Step 3b: conclusion of the proof in case α < 1. We start out with an easier case 0 < α < 1where control over u′ is straightforward from the e-quantity. Indeed, the e-quantity is uniformly bounded by(292), while Λαρ ∈ L∞ simply by |Λαρ|∞ ≤ |ρ′|∞. So, we obtain uniform bound on |u′|∞ and hence globalexistence by Theorem 8.5. However, this argument does not provide a good quantitative estimate on |u′| toconclude flocking. We will seek more precise estimates with the help of non-local maximum principle andfurther fractional estimates.

Let is write the equation for u′, evaluated at maximum of |u′| and multiplied by u′:

(303)d

dt|u′|2 ≤ |u′|3 + u′(x)

∫Rδzu′(x)ρ(x+ z)

dz

|z|1+α+ u′(x)

∫Rδzu(x)ρ′(x+ z)

dz

|z|1+α.

The dissipation term is bounded, as before by

u′(x)

∫Rδzu′(x)ρ(x+ z)

dz

|z|1+α≤ −cDαu

′(x).

For Dαu′(x) we can derive another two versions of the non-local maximum principle similar to (299). First,

replacing instead of u′ with (u− u)′ from the beginning we obtain the amplitude A rather than |u|∞ in thedenominator:

(304) Dαu′(x) ≥ c |u

′(x)|2+α

Aα(t).

And second, by setting r = B−1/α where B is an arbitrary constant, we obtain

Du′(x) ≥ c1B|u′(x)|2 − c2|u′(x)|AB 1+αα ≥ c3B|u′(x)|2 − c4B

2+αα A2.

So, we obtain the following bound for any B > 0:

(305) Dαu′(x) ≥ B|u′(x)|2 − c2B

1+αα A2.

We now continue with the last term in (303). First, let us consider the case 0 < α < 1. Then

u′(x)

∫Rδzu(x)ρ′(x+ z)

dz

|z|1+α≤ u′(x)

∫|z|<1

δzu(x)ρ′(x+ z)dz

|z|1+α

+ u′(x)

∫|z|>1

δzu(x)ρ′(x+ z)dz

|z|1+α

≤ c1|u′(x)|2|ρ′|∞ + c2|u′(x)|A.Picking B > 2c1|ρ′|∞ and using (305) for half of the dissipation term and (304) for the other half, we obtainWe continue

(306)d

dt|u′|2 ≤ |u′|3 − c |u

′(x)|2+α

Aα(t)+ C|u′(x)|A(t) + CA2(t)


We already know from the remark in the beginning of this step that |u′| is uniformly bounded in the case0 < α < 1. So, we estimate |u′|3 . |u′|2 which again gets absorbed in the dissipation in view of (305).Finally,

(307) |u′(x)|A(t) ≤ |u′(x)|2 +A2(t)

which again gets absorbed by cost of adding another A2(t). In the end, we arrive at

d

dt|u′|2 ≤ −c |u

′(x)|2+α

Aα(t)+ CA2(t),

which gives uniform control over |u′|∞, and implies exponential rate of convergence to zero as t → ∞.Since on Step 3a we showed that ρ is uniformly bounded W 1,∞, the proof of strong flocking for the density,ρ → ρ(· − ut), follows along the lines of Theorem 9.1. Lastly, showing exponential decay of |u′′|∞ followssimilar estimates on the evolution of the norm |u′′|2∞, and will not be presented here for the sake of brevity.We refer to [91] for full details.

Step 3c: conclusion of the proof in case α > 1. Next, let us consider the case α > 1. We absorb thecubic term in (303) simply by interpolation

|u′|3 ≤ ε |u′(x)|2+α

Aα(t)+ cεA

3αα−1 (t).

It comes again to estimating the last term in (303). In the long range |z| > 1 we estimate it by |u′(x)|A(t)and treat as before in (307). In the short range we add and subtract u′(x)z. Noting that by (301) we obtain

u′(x)

∫|z|<1

[δzu(x)− u′(x)z]ρ′(x+ z)dz

|z|1+α≤ C|u′|

√Dαu′(x) ≤ εDαu

′(x) + cε|u′|2,

where the first term is absorbed and the quadratic term is treated as before. For the remaining term wehave

|u′(x)|2∫|z|<1

ρ′(x+ z)z dz

|z|1+α= |u′(x)|2

∫|z|>1

ρ′(x+ z)z dz

|z|1+α+ |u′(x)|2

∫Rρ′(x+ z)

z dz

|z|1+α.

The integral over R is nothing other than Λαρ(x) which we replace with e− u′. We obtain

|u′(x)|2∫|z|<1

ρ′(x+ z)z dz

|z|1+α≤ c1|u′(x)|2 + c2|u′(x)|3.

Both terms have been estimated already before. So, we arrive at

d

dt|u′|2 ≤ −c1

|u′(x)|2+α

Aα(t)+ c2Aβ(t), β > 0,

and the result follows.

Step 3d: conclusion of the proof in case α = 1. Case α = 1 is more involved because dissipationis not sufficient to control the non-linearity and the two terms in e, u′ and Λ1ρ, are in balance. So, inorder to proceed we need to establish an additional uniform estimate on the second derivative of ρ in L2,|ρ′′|2 ∈ L∞([0, T0)). This can be done bypassing any additional information about u.

Assuming that we have proved |ρ′′|2 ∈ L∞([0, T0)) with a constant independent of T0, we can concludethe proof the theorem as follows. First, let us denote Λ = Λ1. We establish control over Λρ as follows :

Λρ(x) =

∫|z|<1


|z|2 +

∫1<|z|

δzρ(x)dz

|z|2 .

The second integral is clearly bounded uniformly. Next,

|δzρ(x)− ρ′(x)z| =∣∣∣∣∫ z

0

∫ w

0

ρ′′(x+ y) dy

∣∣∣∣ ≤ |ρ′′|2|z|3/2.So, the first integral is bounded by a constant multiple of |ρ′′|2. This shows that Λρ ∈ L∞([0, T0);L∞). Thisis of course natural because by the Sobolev embedding theorem we even have H2 →W 3/2,∞.

Having uniform control over |Λρ|∞ we immediately obtain control over u′ = e− Λρ and hence the globalexistence follows. But we also note that in this case |u′|3 . |u′|2, and hence by interpolation the cubic termin (303) hides into dissipation by cost of adding a power of A, an exponentially decaying quantity. For the

96 ROMAN SHVYDKOY

last term in (303) we do exact same computation as in the case α > 1, except for the short range part weestimate

|u′(x)|2∫|z|<1

ρ′(x+ z)dz

z= |u′(x)|2

∫|z|<1

[ρ′(x+ z)− ρ′(x)]dz

z

= |u′(x)|2∫|z|<1

∫ x+z

x

ρ′′(w)dz

z

≤ c1|u′(x)|2|ρ′′|2 ≤ c2|u′(x)|2.This concludes the estimate of all terms in (303) we finish the proof as in the previous two case.

So, it remains to obtain a uniform estimate on |ρ′′|2. Let us write the equation for the second derivativeof density:

∂tρ′′ + uρ′′′ + u′ρ′′ + e′′ρ+ 3e′ρ′ + 2eρ′′ =

2ρ′′Λρ+ 3ρ′Λρ′ + ρΛρ′′.(308)

Let us apply the test-function ρ′′/ρ. Via routine computation with the use of the density equation, one canobserve that ⟨

∂tρ′′ + uρ′′′ + u′ρ′′,

ρ′′

ρ

⟩=

1

2∂t

∫1

ρ|ρ′′|2 dx.

In view of the bounds on the density we note that∫

1ρ |ρ′′|2 dx ∼ |ρ′′|22. So, it is sufficient to bound the

rest of the terms in terms of |ρ′′|22. Considering the last three terms on the left hand side, let us make oneobservation: since q′/ρ is transported, then (q′/ρ)′ satisfied the continuity equation, and hence (q′/ρ)′/ρ istransported again. Solving for e′′ in this expression results in poinwise bound

(309) |e′′(x, t)| ≤ C(|ρ′′(x, t)|+ |ρ′(x, t)|+ ρ(x, t)).

In order for this bound to make sense we require m ≥ 3. With the use of a priori estimates established sofar, ⟨

e′′ρ+ 3e′ρ′ + 2eρ′′,ρ′′

ρ

⟩. 1 + |ρ′′|22.

At this point we have (droping ρ, ρ′ that are already bounded)

∂t

∫1

ρ|ρ′′|2 dx . 1 + |ρ′′|22 +

∫|ρ′′|2|Λρ|dx+

+

∫|ρ′′||Λρ′|dx+

∫ρ′′Λρ′′ dx

= 1 + |ρ′′|22 + I1 + I2 + I3.

(310)

Clearly, the last term I3 is dissipative:

I3 . −∫

Dαρ′′(x) dx− 1

|ρ′|∞

∫|ρ′′|3 dx = −‖ρ′′‖2

H12− |ρ′′|33,

where in the latter we dropped 1|ρ′|∞ from inside the integral since this term is bounded from below.

To tackls I1 we fix ε > 0 small and split the fractional Laplacian:

Λρ(x) =

∫|z|<ε


|z|2 +

∫ε<|z|

δzρ(x)dz

|z|2 ≤ ε1/2|ρ′′|2 + cε.

So,

I1 ≤ ε1/2|ρ′′|32 + cε|ρ′′|22 ≤ ε1/2|ρ′′|33 + cε|ρ′′|22.The cubic term gets absorbed by dissipation for small ε.

For I2 we simply use the Holder inequality:

|I2| ≤ |ρ′′|2|Λρ′|2 . |ρ′′|22.So,

(311) ∂t

∫1

ρ|ρ′′|2 dx . 1 + |ρ′′|22 − |ρ′′|33.


This finishes the proof.

A few remarks are in order. First, the global existence part of the theorem holds for local kernels as well.In other words if we only know that

φ(z) ∼ 1

|z|1+α, for |z| < r0,

the proof goes through as long as we can establish bounds on the density from above and below. The boundfrom above follows the steps of the proof in the global kernels. Indeed, we only need singularity in shortrange to be able to pick small parameter r to achieve (290). As to the bound from below, unfortunatelywe loose much more. However, we can still find an algebraic bound from below for any non-negative kernel!Indeed, arguing as in (291) we simply drop the integral term and arrive at

d

dtρ− ≥ −c1ρ2

−,

which implies

(312) ρ(x, t) &1

1 + t, ∀φ ≥ 0.

This, of course, is not sufficient to establish exponential alignment in the local case, but is sufficient to applythe continuation argument. We do however recover weak flocking as a consequence of Theorem 9.4. Let usrecord these observations in the following theorem.

Theorem 9.10. Suppose the kernel is given by

φ(z) =h(|z|)|z|1+α

, 0 < α < 2.

Suppose for some m ≥ 3, (u0, ρ0) ∈ Hm+1(T) ×Hm+α(T) and ρ0(x) > 0 for all x ∈ Tn. Then there existsa unique non-vacuous global in time solution to (243) in the class

(313) u ∈ Cw([0,∞);Hm+1) ∩ L2loc([0,∞); Hm+1+α/2), ρ ∈ Cw([0,∞);Hm+α).

Moreover, the solution obeys bounds on the density

(314)c0

1 + t≤ ρ(x, t) ≤ C0, t ≥ 0,

and weak alignment V2(t)→ 0, as t→∞.

9.5. Notes and References. Critical threshold conditions for global regularity of Euler alignment modelsfirst appeared in Tan and Tadmor work [94], although the sharp criterion of Theorem 9.1 was found later inCarrillo et al [16]. The concept of strong flocking (248) was first introduced and proved actually for singularmodels on T in [91, 92], and in the open space in the context of smooth multi-scale models in [88]. We havepresented the adapted proof for mono-scale case.

The role of the e-quantity as an entropy and Theorem 9.2 was proved in Leslie, Shvydkoy [66]. Thecases of singular and topological models were also included in that study. The corrector method on T andTheorem 9.4 was proved in [38].

Singular models and proof of Theorem 9.8 first appeared in Shvydkoy, Tadmor [90] for the case α ≥ 1, andshortly after in Do, Kiselev, Ryzhik, and Tan for the case 0 < α < 1. Later the whole range of α together withstrong flocking was covered in [91, 92]. The approach presented here was taken from [90]. The work of Doet al relies on an adaptation of the modulus of continuity method used to make several breakthroughs in theregularity theory of critical fractional diffusion equations such as critical SQG by Kiselev, Nazarov, Volberg[59]. The latter enjoyed several other different proofs using DiGiorgi method by Caffarelli and Vasseur [11]and the nonlocal maximum principle by Constantin and Vicol [29] which we also used in our proof. Aninteresting extension of Theorem 9.8 as well as Holder regularization to the case α = 0 was obtained in Anand Ryzhik [2]. Other extensions include construction of weak and strong solutions for singular model withexternal force by Leslie [65], and global existence of classical solutions with potential attraction/repulsionforces by Kiselev and Tan [60].

98 ROMAN SHVYDKOY

Regularity of topological models with singular kernels given by

φ(x, y) =h(x− y)

dτ (x, y)|x− y|1+α−τ

was developed in [89] with the use of a blend of techniques including DiGiorgi method, fractional Schauderestimates and Silvestre regularization results [93] depending on the range of α. The final result states globalexistence in Sobolev classes provided τ ≤ α or if τ > α then under an additional smallness condition. Theτ ≤ α condition is needed to prove global a priori bounds on the density, whereby the metric component ofthe kernel plays a decisive role.


10. Global solutions to multi-dimensional systems

Global existence of Euler alignment system in dimension 2 and higher is an open problem. Only partialanswers are known for smooth models and small initial data results for singular models (see also Theorem 8.3).In this chapter we will describe some special classes of multi-dimensional solutions and present two kinds ofsmall initial data results.

10.1. Unidirectional flocks and their stability. One class of solutions that behaves like 1D is the classof unidirectional oriented flows. These are given by

(315) u(x, t) = u(x, t) d, d ∈ Sn−1, u : Rn × R+ → R.The same conservation law holds for the entropy

e = d · ∇u+ φ ∗ ρ, ∂te+∇ · (eu) = 0,

d

Figure 13. Oriented flow

although in this case the entropy does not control the full gradi-ent of the velocity field. Nonetheless, one can develop a theoryfully analogous to the 1D in the smooth communication case,which is what we will cover in this section.

First of all by the maximum principle applied in any direc-tion perpendicular to d one can see that the ansatz (315) ispreserved in time. Second, in view of rotational invariance ofthe Euler Alignment System, we can assume that d points inthe direction of the x1-axis. So, we can assume

(316) u(x, t) = 〈u(x, t), 0, . . . , 0〉 for u : Rn × R+ → R.The full system (169) takes for of a system of scalar conserva-

tion laws:

(317) (x, t) ∈ Rn × R+

∂tρ+ ∂1(ρu) = 0,∂tu+ 1

2∂1(u2) = φ ∗ (ρu)− uφ ∗ ρ.The entropy takes form

(318) e := ∂1u+ φ ∗ ρ, ∂te+ ∂1(ue) = 0.

As a result global well-posedness follows in the same way as in 1D via the threshold condition e0 ≥ 0. Wecan prove the full analogue of the one dimensional Theorem 9.1:

(319) A(t) + ‖∇u(t)‖L∞(Supp(ρ(t)) + ‖∇2u(t)‖L∞(Supp(ρ(t)) ≤ Ce−δt,together with strong flocking (248).

We start as before by noting that the diameter of the flock D(t) remains bounded. So, there exists a timet? > 0 such that e(x, t) ≥ 1

2φ(D)M for all x ∈ Supp ρ(·, t) and t > t?. Let us write an equation for ∂iu inthe following form

∂t∂iu+ ∂iu∂1u+ u∂i∂1u = ∂iφ ∗ (ρu)− ∂iuφ ∗ (ρ)− u∂iφ ∗ (ρ),

or along characteristics

d

dt∂iu = C∂iφ(u, ρ)− e ∂iu.(320)

Recall from Theorem 7.1 that the velocity fluctuations A(t) are exponentially decaying. Hence, the integralabove will be bounded by |∂iφ|∞ME(t) where we denote by E(t) a generic exponentially decaying quantity.Evaluating (320) at the maximum over Supp ρ(·, t) we obtain

∂t‖∂iu‖L∞(Supp ρ(·,t)) ≤ E(t)− 12φ(D)M‖∂iu‖L∞(Supp ρ(·,t)).

This readily implies the exponential bound on ‖∂iu‖L∞(Supp ρ(·,t)) for i = 1, . . . , n.Moving on to the second order derivatives, we write the equations along characteristics

d

dt∂j∂iu =

∫Ω

∂j∂iφ(|x− y|) (u(y)− u(x)) ρ(y) dy − e ∂j∂iu− ∂je ∂iu− ∂ie ∂ju,

where∂je = ∂j∂1u+ ∂jφ ∗ ρ and ∂ie = ∂i∂1u+ ∂iφ ∗ ρ.

100 ROMAN SHVYDKOY

We prove exponential decay by bootstrapping information from the partial ∂1∂1, then ∂1∂j , and thengeneral ∂i∂j . So, first, we consider the case i = j = 1. Note that for this particular case, we have

d

dt∂2

1u =

∫Ω

∂21φ(|x− y|) (u(y)− u(x)) ρ(y) dy − e ∂2

1u− 2 ∂1e ∂1u.

Using that ∂1e = ∂21u+ ∂1φ ∗ ρ, we arrive at

d

dt∂2

1u ≤ E(t)− ∂21u(e− E(t)).

Now, as e(x, t) ≥ 12φ(D)M > 0 for all x ∈ Supp ρ(·, t) and t > t? we have that

d

dt∂2

1u ≤ E(t)− ∂21u(

12φ(D)M − E(t)

)for t > t?.

Since E(t) decays exponentially fast, there must exists t?? > t? such that

d

dt∂2

1u ≤ E(t)− 14φ(D)M∂2

1u for t > t??.

Then, evaluating the previous inequality at the maximum over Supp ρ(·, t) we have obtained the desiredresult by integration.

Second, we consider the case i = 1 and j 6= 1. In this case, we have

d

dt∂j∂1u =

∫Ω

∂j∂1φ(|x− y|) (u(y)− u(x)) ρ(y) dy − e ∂j∂1u− ∂je ∂1u− ∂1e ∂ju,

where

∂je = ∂j∂1u+ ∂jφ ∗ (ρ) and ∂1e = ∂21u+ ∂1φ ∗ (ρ).

Using that ‖∇u‖L∞(Supp ρ(·,t)) ≤ E(t) and the fact that ‖∂21u‖L∞(Supp ρ(·,t)) ≤ E(t) we get

d

dt∂j∂1u ≤ E(t)− ∂j∂1u (e− E(t))

and doing the same as before we obtain that ‖∂j∂1u‖L∞(Supp ρ(·,t)) ≤ E(t) for j 6= 1.Finally, the case i, j 6= 1 relies on the previous in a similar manner. We get

d

dt∂j∂iu ≤ E(t)− e ∂j∂iu

and hence ‖∂j∂iu‖L∞(Supp ρ(·,t)) ≤ E(t).The same argument as in 1D shows that |∇ρ|∞ remains uniformly bounded, and with the exponential

decay of velocity this implies strong flocking by the same argument as in 1D.Remarkably any 2D-perturbation of a unidirectional flow is still globally well-posed and remains small. It

is surprising because such perturbations do not obey the same 1D entropy conservation law for the e-quantity.Yet, their existence and stability can be established.

Theorem 10.1. Consider the Euler Alignment system (169) with smooth kernel φ on the periodic domainTn. Let (u0, ρ0) ∈ Hm × (L1

+ ∩W k,∞) with m ≥ k + 1 > n2 + 2 and the initial velocity has form

(321) u0(x) = u0(x)d + v0(x)d? for some d,d? ∈ Sn−1,

satisfying

(322) infx∈Tn

e0(x) ≥ √ε, ‖u0‖W 1,∞ . 1, ‖v0‖W 1,∞ . ε2,

for ε small enough. Then there exists a global in time solution to (169) which is stable around the underlyingunidirectional motion:

(323) |∇v(t)|∞ . ε, ∀t > 0.

The rest of this section will be devoted to the proof of this statement. The key idea is to analyze anevolution equation for the whole expression on the right hand side of the e-equation and to establish controlover its magnitude which we denote

R(t) := ‖(∇ · u)2 − Tr[(∇u)2]‖L∞ .


We observe that initially R(0) . ε2 and by continuity R(t) . ε2 at least for a short period of time. So, letus define a possible critical time t? at which the solution hypothetically reaches size ε for the first time:

R(t∗) = ε, R(t) < ε for t < t?.

A contradiction will follows if we establish that R′(t?) < 0. This would imply the bound R(t) < ε on theentire interval of existence, which in turn will imply a bound on e, and since φ ∗ ρ is bounded a priori, weconclude a bound on ∇ · u. This would imply global existence due to Theorem 8.1.

So, let write the equation for (∇ ·u)2−Tr[(∇u)2]. First, due to rotational invariance of the system (169)one can assume for simplicity that d points in the direction of the x1-axis and d? points in the direction ofthe x2-axis:

(324) u0(x) = 〈u0(x), v0(x), 0, . . . , 0〉.Then by the maximum principle the solution will remain two dimensional at all time:

(325) u(x, t) = 〈u(x, t), v(x, t), 0, . . . , 0〉.Let us define a Poisson-type bracket

f, Cφ(g, h) := ∂1f C∂2φ(g, h)− ∂2f C∂1φ(g, h),

and denote

Υφ(u, v) := 2 u, Cφ(ρ, v)+ 2 v, Cφ(ρ, u) .Directly from the Euler Alignment system we obtain the following equation with time derivative being

along characteristics of u:

(326)d

dt

[(∇ · u)2 − Tr[(∇u)2]

]= − (e+ (φ ∗ ρ))

[(∇ · u)2 − Tr[(∇u)2]

]+ Υφ(u, v).

Lemma 10.2. We have the following bounds on the time interval [0, t∗]:

(327) 12

√ε ≤ e(x, t) ≤ C0 := 2 max|e0|∞,M|φ|∞.

Proof. Indeed, let us recall

(328)d

dte =

[(∇ · u)2 − Tr[(∇u)2]

]+ e ((φ ∗ ρ)− e) .

If e(x, t) = 12

√ε for the first time t < t∗, then

d

dte ≥ −ε+ 1

2

√ε(φ ∗ ρ(x, t)− 1

2

√ε) ≥ −ε+ 1

2

√ε(M inf

xφ− 1

2

√ε) > 0,

provided ε .M2, a contradiction. At the same time, if e(x, t) = C0 > |e0|∞, then

d

dte ≤ ε+ C0(|φ|∞M− C0) < 0,

provided C0 > 2M|φ|∞. So, with the choice of C0 = 2 max|e0|∞,M|φ|∞ we have proved the lemma.

Next, we establish control over the partial gradient |∇1,2u(t)|∞ which is needed to bound the residualterm Υφ(u, v) in (326). We write an equation for the full gradient in the form suitable for our analysis. Wehave for the general solutions u = (u1, . . . , un),

∂t∂luk + (∂lu · ∇)uk + (u · ∇)∂lu

k = C∂lφ(ρ, uk)− ∂luk(φ ∗ ρ).

Therefore, we obtain that

d

dt∂lu

k = C∂lφ(ρ, uk)− ∂luk(φ ∗ ρ)− (∂lw · ∇)uk

= C∂lφ(ρ, uk)− e ∂luk + ∂luk(∇ · u)− (∂lu · ∇)uk,

where

∂luk(∇ · u)− (∂lu · ∇)uk =

n∑i=1

∣∣∣∣∂iui ∂lui

∂iuk ∂lu

k

∣∣∣∣ .

102 ROMAN SHVYDKOY

We therefore arrive at

(329)d

dt∂lu

k −n∑i=1

∣∣∣∣∂iui ∂lui

∂iuk ∂lu

k

∣∣∣∣ = C∂lφ(ρ, uk)− e ∂luk.

For our solutions with only two non-zero components the gradient takes form

∇u =

∂1u ∂2u∂1v ∂2v

∂3u . . . ∂nu∂3v . . . ∂nv

0 0...

...0 0

0 . . . 0...

. . ....

0 . . . 0

≡(P Q0 0

)

We first address the upper corner part P .

Lemma 10.3. We have the following bounds on the time interval [0, t∗]

(330) |∂1v(t)|∞ + |∂2v(t)|∞ ≤ Cε3/2, |∂1u(t)|∞ + |∂2u(t)|∞ ≤ Cε−1/2.

Proof. First, let us consider the off-diagonal entries of P :

(331)

d

dt∂2u = C∂2φ(ρ, u)− e ∂2u,

d

dt∂1v = C∂1φ(ρ, v)− e ∂1v.

In view of (327) we have

∂t|∂1v|∞ ≤ 2 |∇φ|∞M |v0|∞ −1

2

√ε |∂1v|∞,

∂t|∂2u|∞ ≤ 2 |∇φ|∞M |u0|∞ −1

2

√ε |∂2u|∞.

By Gronwall’s inequality

|∂1v|∞ ≤ |∂1v0|∞ e−12

√εt +

2√ε|∂1φ|∞M|v0|∞(1− e− 1

2

√εt) ≤ Cε3/2

|∂2u|∞ ≤ |∂2u0|∞ e−12

√εt +

2√ε|∂2φ|∞M|u0|∞(1− e− 1

2

√εt) ≤ Cε−1/2.

As to the diagonal entries of P , we have

(332)

d

dt∂1u− 1

2

[(∇ · u)2 − Tr[(∇u)2]

]= C∂1φ(ρ, u)− e ∂1u,

d

dt∂2v − 1

2

[(∇ · u)2 − Tr[(∇u)2]

]= C∂2φ(ρ, v)− e ∂2v.

Thus, on the interval [0, t∗]:

∂t|∂1u|∞ ≤ ε+ 2 |∂1φ|∞M |u0|∞ −1

2

√ε |∂1u|∞.

Consequently,

|∂1u|∞ ≤ |∂1u0|∞ e−12

√εt +

2√ε

(|∂1φ|∞M |u0|∞ + ε)(1− e− 12

√εt) ≤ Cε−1/2.

A similar estimate on ∂2v gives us . ε1/2, which is not sufficient. Instead, we rewrite the equation for ∂2vin the following way:

(333)d

dt∂2v + ∂2u ∂1v + ∂2v∂2v = C∂2φ(ρ, v)− (φ ∗ ρ) ∂2v.

We define the points x±(t) where the maximum and respectively minimum of ∂2v(x, t) is attained. Then,

d

dt∂2v(x+(t), t) ≤ −

[(φ ∗ ρ) + ∂2v(x+(t), t)

]∂2v(x+(t), t) + Cε3/2,

−d

dt∂2v(x−(t), t) ≤

[(φ ∗ ρ) + ∂2v(x−(t), t)

]∂2v(x−(t), t) + Cε3/2.


So, the difference d(t) := ∂2vε(x+(t), t)− ∂2v

ε(x−(t), t) satisfies

(334) d′(t) ≤ −[(φ ∗ ρ) + ∂2v

ε(x+(t), t)− ∂2vε(x−(t), t)

]d(t) + Cε3/2.

We already established that |∂2v|∞ ≤ ε1/2. Using this we obtain

(φ ∗ ρ) + ∂2vε(x+(t), t)− ∂2v

ε(x−(t), t) ≥ c0,and hence

d′(t) ≤ −c0d(t) + Cε3/2, d(0) ≤ ε2.

Another application of Gronwall’s lemma gives d(t) . ε3/2 and the proof is complete.

With these ingredients at hand we now look back at the equation (326). At the critical time t?, we have

d

dtR(t?) ≤ −cMR(t?) + Υφ(u, v)

We have, using (330) and |v|∞ ≤ |v0|∞ ≈ ε2,

u, Cφ(ρ, v) ≤ Cε−1/2ε2 = Cε3/2

v, Cφ(ρ, u) ≤ Cε3/2.

Consequently,d

dtR(t?) ≤ −cε+ Cε3/2 < 0

for ε > 0 small enough. This shows that t∗ =∞, and hence by Lemma 10.2 we have a uniform bound on e,which fulfills the continuation criterion of Theorem 8.1.

Lastly, we establish control over the remaining part Q of the gradient matrix.

Lemma 10.4. We have for all k = 3, . . . , n and all time t > 0,

|∂ku(t)|∞ ≤ cε−1/2, |∂kv(t)|∞ ≤ cε.Proof. Let us write the system for ∂ku and ∂kv:

d

dt∂ku− ∂2v ∂ku+ ∂2u ∂kv = C∂kφ(ρ, u)− e ∂ku,

d

dt∂kv + ∂1v ∂ku+ ∂2v ∂kv = C∂kφ(ρ, v)− (φ ∗ ρ) ∂kv.

(335)

Denote Xk(t) := |∂ku(t)|∞ and Yk(t) := |∂kv(t)|∞ for k = 3, . . . , n. Combining (330) together with the lowerbound (φ ∗ ρ) ≥ c0 and the fact that e(t) ≥ 1

2

√ε, we obtain

(336)

Xk(t) ≤ −c1√εXk(t) +

c2√εYk(t) + c3,

Yk(t) ≤ c4ε3/2Xk(t)− c5Yk(t) + c6ε2,

with all the constants being independent of ε. Defining the vector Zk(t) := (Xk(t), Yk(t)) the system (336)can be rewritten in matrix form

Zk(t) ≤ AZk(t) + b

with diagonalization A = PDP−1, where

A :=

(−c1√ε c2√ε

c4ε3/2 −c5

), b :=

(c3c6ε

2

)and D :=

(λ+(ε) 0

0 λ−(ε)

).

Noting that Tr(A) < 0 and Det(A) > 0 for ε > 0 small enough, both eigenvalues λ±(ε) are negative. Morespecifically, we have

λ±(ε) := Tr(A)2 ±

√(Tr(A)

2

)2

− det(A)

withλ+(ε)→ −c5 +O(ε) and λ+(ε)→ −√ε+O(ε).

By Duhamel’s formula,

Zk(t) ≤ eAtZk(0) +

∫ t

0

eA(t−s)b ds.

104 ROMAN SHVYDKOY

r1

r2

bb

b

b

b

rA

b

Figure 14. Mikado cluster in v-variables.

So, calculating eAt = PeDtP−1 leads to the solution to the system, by simply integrating with respect to t.Using that Zk(0) ≈ (1, ε2) and elementary linear algebra, we obtain

Xk(t) .(eλ−t + eλ+t

)+

(eλ−t − 1

λ−+eλ+t − 1

λ+

),

Yk(t) . ε3/2(eλ−t − eλ+t

)+ ε3/2

(eλ−t − 1

λ−− eλ+t − 1

λ+

).

Combining the above we conclude

Xk(t) ≤ c7ε−1/2, Yk(t) ≤ c8ε, ∀t ≥ 0,

as desired.

Together with the previously established bound (330) we obtain |∇v(t)|∞ . ε. This finishes the proof ofTheorem 10.1.

10.2. Mikado clusters in hydrodynamic multi-flocks. The macroscopic counterpart of the discretemulti-flock system (57) introduced in Section 3.8 can be derived in a similar fashion to the mono-flock case.Letting macroscopic flock variables denoted by (ρα,uα) for α = 1, . . . , A and global flock parameters by

Xα(t) :=1

Mα

∫Rnx ρα(x, t) dx, Mα :=

∫Rnρα(x, t) dx, Vα(t) :=

1

Mα

∫Rn

uα(x, t)ρα(x, t) dx,

the multiflock Euler Alignment system is given by

(337)

∂tρα +∇ · (ραuα) = 0,

∂tuα + uα · ∇uα = λα[φα ∗ (ραuα)− uα (φα ∗ ρα)] + ε∑β 6=α

MβΨ(Xα −Xβ) (Vβ − uα) ,

where as before communication between flocks is assumed to be weaker than communication inside each ofthe flocks ε minα λα.

One can develop a similar regularity theory in 1D as for mono-flock case and show analogues of Theo-rem 3.12 and Theorem 3.13 together with strong flocking statements. Here the α-entropies eα = ∂xuα +λαφα ∗ ρα determine the threshold condition eα ≥ 0 for global existence, see [88].

Now, by analogy with the discrete system we can pass the new system of variables tied to the referenceframe of each flock

vα(x, t) := uα(x−Xα(t), t)−Vα(t) and %α(x, t) := ρα(x−Xα(t), t).

The new system reads

(338)

∂t%α +∇ · (%αvα) = 0,

∂tvα + vα · ∇vα = λα [φα ∗ (%αvα)− vα(φα ∗ %α)] + εRα(t)vα,


where

Rα(t) :=∑β 6=α

MβΨ(Xα(t)−Xβ(t)).

Now, each flock satisfies the maximum principle and we can study unidirectional configurations:

(339) vα(x, t) = vα(x, t) rα for vα : Rn × R+ → R, rα ∈ Sn−1,

We call these solutions Mikado clusters, see Figure 14 – by analogy with Mikado solutions to the 3D in-compressible Euler equation which played crucial role in resolution of the celebrated Onsager conjecture,[36, ?].

Theorem 10.5. Consider initial Mikado cluster (339) with (vα(0), ρα(0)) ∈ Hm × (L1+ ∩ W 1,∞) with

m > n2 +2, satisfying the threshold condition eα(0) ≥ 0 for all α = 1, . . . , A. Then there exists a global in time

unique solution to system (337) which retains the same form (339) and satisfies (vα, ρα) ∈ Cw([0,∞);Hm×(L1

+ ∩W 1,∞). Moreover,

• [Fast local flocking]. Assuming that for a given α ∈ 1, . . . , A the α-flock has compact support andthe internal kernel φα has a heavy tail, then there exists δα(λα, φα, ρα(0),uα(0)) such that

supx∈Supp ρα(·,t)

[|uα(x, t)−Vα(t)|+ |∇uα(x, t)|+ |∇2uα(x, t)|

]. e−δαt,

|ρα(·, t)− ρα(· −Xα(t))|Cγ . e−δαt (0 < γ < 1).

• [Slow global flocking]. Suppose the inter-flock kernel Ψ has a heavy tail and the internal kernelsφα ≥ 0 are arbitrary. If the multi-flock has a finite diameter initially, then global alignment occursat a rate δ(Ψ, ε, ρα(0),uα(0)) such that

supx∈Supp ρα(·,t)α=1,...,A

[|uα(x, t)−V|+ |∇uα(x, t)|+ |∇2uα(x, t)|

]. e−δt,

|ρα(·, t)− ρα(· − tV)|Cγ . e−δt (0 < γ < 1),

where V = 1M

∑Aα=1MαVα is the global momentum.

As in the monoflock case the entropy plays a crucial role,

(340) eα := rα · ∇vα + λαφα ∗ %α,which satisfies

∂teα +∇ · (vα%α) = −εRα(t) (∇ · vα),

or equivalently along characteristics

d

dteα = (εRα(t) + eα)(λαφα ∗ %α − eα).

Since Rα ≥ 0, the initial positive entropy eα ≥ 0 will preserve its sign, and also be globally bounded. Thus,∇ · uα is bounded, and hence we obtain global existence by Theorem 8.1.

It was already shown in [88] that any classical solution to a multi-flock aligns exponentially fast. To provestrong flocking we simply observe that the scalar pair (vα, %α) satisfies

(341)

∂t%α +∇ · (%αvαrα) = 0,

∂tvα + (rα · ∇vα) vα = λα [φα ∗ (%αvα)− vα(φα ∗ %α)] + εRα(t)vα,

which is similar to (317) with the exception of the extra term εRα(t)vα, which is a damping term sinceRα ≥ 0. So, the same analysis as in the previous section applies.

10.3. Spectral dynamics approach. In dimension 2 one can obtain an alternative threshold conditionbased on spectral dynamics approach. Let us assume that the kernel φ is smooth, of convolution type, andsatisfies the usual heavy tail condition (8). Recall the entropy

(342) e = ∇ · u + φ ∗ ρ,satisfying the equation

(343) et +∇ · (ue) = (∇ · u)2 − Tr(∇u)2.

106 ROMAN SHVYDKOY

In 2D the right hand side is equal exactly to 2 det(∇u). So, if we attempt to appeal as in 1D to the logisticnature of the equation we write

d

dte = e(φ ∗ ρ− e) + 2 det(∇u).

So, the residual term det(∇u) gets in the way of controlling the growth or sign of e. It is difficult howeverto track down dynamics of det(∇u) since ∇u is non-symmetric. Instead, one can track the dynamics of thesymmetric part of ∇u, and in particular the eigenvalues of S = 1

2 (∇u +∇⊥u). In order to see exactly whatwe are aiming for, let us note that

det(∇u) = detS + ω2,

where ω = 12 (∂1u2 − ∂2u1) is the scalar vorticity of the field. Denote by µ1, µ2 the eigenvalues of S. Then

detS = µ1µ2. At the same time

µ1µ2 =1

4(µ1 + µ2)2 − 1

4(µ1 − µ2)2.

The first term is exactly 14 (∇ · u)2, and the second involves spectral gap denoted η = µ1 − µ2. So,

2 det(∇u) =1

2(∇ · u)2 − 1

2η2 + 2ω2.

Expanding ∇ · u = e− φ ∗ ρ the e-equation can now be rewritten as follows

(344) 2d

dte = (φ ∗ ρ)2 + 4ω2 − η2 − e2.

The issue now reduces to whether we can control the spectral gap η and vorticity ω. It turns out thatevolution of both quantities can be read off easily from the equation for ∇u. Indeed, let us write the fullmatrix equation first:

(345) ∂t∇u + u · ∇(∇u) + (∇u)2 = −(φ ∗ ρ)∇u + E,

where E is an exponentially decaying quantity. To be precise,

E = C∇φ(u, ρ),

and according to (25),

(346) |E| ≤ A0e−λMφ(D)t|∇φ|∞M,

where D is determined solely from initial condition by equation (28). We decompose ∇u into symmetric andskew-symmetric parts

∇u = S + Ω, S =1

2(∇u +∇u>), Ω =

(0 −ωω 0

).

And we have decomposition of the square matrix

(∇u)2 = S2 − ω2I2×2︸︷︷︸sym

+SΩ + ΩS︸︷︷︸skew−sym

= S2 − ω2I2×2 + Ω∇ · u.

So, reading off the equation for skew-symmetric part we obtain (in Lagrangian coordinates)

d

dtω + eω = E.

For the symmetric part we have

(347)d

dtS + S2 = ω2I2×2 − (φ ∗ ρ)S + E.

Now the advantage of considering symmetric S come into light at this point. Denote (s1(x, t), s2(x, t)) theorthonormal basis of eigenvectors corresponding to µ1 and µ2, respectively. Then µi = siSsi, and note that

d

dtµi = si

[d

dtS

]si,


due to orthogonality si · ddtsi = 0. So, multiplying the S-equation by si from both sides, we obtain a system

for spectral dynamics of the eigenvalues

d

dtµi + µ2

i = ω2 − (φ ∗ ρ)µi + E.

Lastly, taking the difference and using that µ21 − µ2

2 = η(∇ · u) we obtain

d

dtη + eη = E.

Collecting together the equations we have obtained the system

(348)

2e+ e2 = (φ ∗ ρ)2 + 4ω2 − η2

ω + eω = E

η + eη = E.

Let us note in passing that the bound on E’s in (346) is still valid up to an absolute constant due to algebraicmanipulations above.

So, let us now fix an initial condition (u0, ρ0) ∈ Hm × (L1+ ∩W k,∞) and assume that e0(x) > 0 for every

x ∈ R2. According to Theorem 8.2 we have a local solution on a maximal time interval [0, T0). By continuity,e(X(t, x), t) > 0 for some short time t < T (x). On that time interval the spectral gap solution reads

η(t) = η0 exp

−∫ t

0

e(s) ds

+

∫ t

0

exp

−∫ t

s

e(τ) dτ

E(s) ds.

So,

|η(t)| ≤ |η0|+ cA0|∇φ|∞φ(D)

.

Using this we obtain from the e-equation

2e ≥ φ2(D)M2 −(|η0|+ cA0

|∇φ|∞φ(D)

)2

− e2.

Assuming now the small gap and small amplitude condition

|η0| <1

4φ(D)M, A0 <

1

4

φ2(D)M

c|∇φ|∞we find that

2e ≥ 1

2φ2(D)M2 − e2.

This shows that e will remain positive on the entire interval [0, T0), which implies that ∇·u remains boundedfrom below. The continuation criterion (223) proves global existence.

Theorem 10.6. Suppose (u0, ρ0) ∈ Hm×(L1+∩W 1,∞) and assume that e0(x) > 0 for every x ∈ R2. Assume

also the smallness conditions

|η0| <1

4φ(D)M, A0 <

φ2(D)M

c|∇φ|∞.

Then there exists a global solution with this initial condition.

It is interesting to note that the small amplitude condition alone would guarantee global existence forsingular models due to additional dissipation enhancement effect, see Section 10.4.

Let us observe that in the case of unidirectional flocks described in the previous section, |η0| = |∇u0|. So,the spectral gap not always applies to those flows. Yet, we know they are globally well-posed. It would beinteresting to bridge the gap between the two classes of solutions.

108 ROMAN SHVYDKOY

10.4. Nearly aligned flocks of singular models: small initial data. Lack of control on e in multidimensional case is part of the reason why the model has no well developed global regularity theory. Forsingular models, however, dissipation provided by the alignment may be a tool to preserve regularity in somemulti dimensional situations. The key mechanism to control the entropy and subsequently obtain a globalexistence of solutions with small initial fluctuations is provided by a non-linear bound on the dissipationterm introduced by Constantin in Vicol in the course of showing global existence for critical SQG equation,[29]. We extend this bound to finite differences of higher order and carry out the small data result in thecase of n-dimensional torus and no vacuum condition.

To fix the notation, we denote by [·]s the metric of the homogeneous Holder class Cs(Tn). For higher s,we will resort to a finite difference definition of [·]s stated as follows. First we denote

δhf(x) = f(x+ h)− f(x), τzf(x) = f(x+ z)

δ2hf = δh(δhf), δ3

hf = δh(δh(δhf)).

We then define, for 0 < γ < 1:

(349) [f ]2+γ = supx,h∈Tn

|δ3hf(x)||h|2+γ

.

The equivalence of (349) to the classical norm [∇2f ]γ is a well known result in approximation theory, see[96]. For integer values of smoothness parameter k ∈ N we use classical homogeneous metric [f ]k = |∇kf |∞.

We consider the global singular communication given by kernel of the classical fractional Laplacian Λα:

(350) φ(z) =∑k∈Zn

1

|z + 2πk|n+α, 0 < α < 2.

Like in 1D it is sometimes convenient to use both open space and periodic representations of Λα:

(351) Λαf(x) = p.v.

∫Tnφ(z)δzf(x) dz = p.v.

∫Rnδzf(x)

dz

|z|n+α.

Let us state the result.

Theorem 10.7. Consider the Euler Alignment System (169) on the torus Tn with kernel given by (350).There exists an N ∈ N such that for any sufficiently large R > 0, depending only on α and dimension n,any initial condition (u0, ρ0) ∈ Hm+1(Tn)×Hm+α(Tn), m > n

2 + 3, satisfying

|ρ0|∞, |ρ−10 |∞, [u0]3, [ρ0]3 ≤ R,

A0 ≤1

RN,

(352)

gives rise to a unique global solution in class Cw([0,∞);Hm+1 ×Hm+α). Moreover, the solution convergesto a flocking state exponentially fast:

A(t) + [u(t)]1 + [u(t)]2 + ‖ρ(t)− ρ(t)‖C1 < Ce−δt.

in the course of the proof of Theorem 10.7 we establish a uniform control on C2-norm of u and the distancebetween the initial density ρ0 and its final profile ρ. As a byproduct, we obtain the following stability resultfor flocking states.

Theorem 10.8. Let (u, ρ) be a traveling wave, where ρ(x, t) = ρ(x− tu), and let (v0, r0) be an initial datasatisfying the conditions of Theorem 10.7. Suppose |v0 − u|∞ + |r0 − ρ0|∞ < ε. Then the solution willconverge to another flock r with |r − ρ0|∞ < εθ, where θ ∈ (0, 1) depends only on α.

The idea of the proof is to establish control over a higher Holder norm [u]2+γ . This serves multiplepurposes. First, it automatically shows boundedness of the gradients |∇u|∞ and |∇ρ|∞, where for thelatter we need to control |∇2u|∞. So, we fulfill the continuation criterion of Theorem 8.5 and concludeglobal existence as stated. Second, with C2+γ norm uniformly bounded, we obtain exponential decay of[u(t)]1 + [u(t)]2 simply by interpolation with A, which readily implies strong flocking as in Theorem 9.1.

From now on we will fix an exponent 0 < γ < 1 to be identified later but dependent only on α.The proof will be structured in several steps.


Step 1: Breakthrough scenario. According to Theorem 8.5 we have a local solution (u, ρ) ∈ Cw([0, T0) :Hm+1×Hm+α) satisfying the assumptions of Theorem 10.7. Note that in view of the smallness assumptionon A0, the norm [u(t)]2+γ will remain smaller than 1 at least for a short period of time. If the solutioncannot be extended beyond T0 there exists a possible critical time t∗ < T0 at which the solution reaches sizeR for the first time:

(353) [u(t∗)]2+γ = R, [u(t)]2+γ < R, t < t∗.

A contradiction is be achieved when we show that ∂t[u(t∗)]2+γ < 0. This establishes the bound [u(t)]2+γ < Ron the entire interval [0, T0), and hence, continuation of the solution beyond T0 by Theorem 8.5. In the courseof the argument we pick γ based on several occurring restrictions, but ultimately depending only on α.

Step 2: Preliminary estimates on [0, t∗]. We will make a few preliminary estimates on various Holdernorms of the data. We fix R and N are sufficiently large, and N depending only on α, for all the argumentsbelow to go through.

First, we notice two direct bounds:

(354) [u(t)]1, [u(t)]2 < R−6α e−

c0tR , for all t ≤ t∗.

Indeed, by interpolation and in view of (352),

[u]1 ≤ A1+γ2+γ

0 [u]1

2+γ

2+γ ≤ R1−N/2e−c0t/R < R−6α e−

c0tR ,

and similarly,

[u]2 ≤ Aγ

2+γ

0 [u]2

2+γ

2+γ < R1−N γ2+γ e−c0t/R ≤ R− 6

α e−c0t/R.

Next, we consider density. Let us denote ρ and ρ the minimum and maximum of ρ, respectively. Denoted = ∇ · u. From (221) we conclude the bounds

ρ0

exp

−∫ t

0

|d(s)|∞ ds

≤ ρ(t), ρ(t) ≤ ρ0 exp

∫ t

0

|d(s)|∞ ds

.

By (354), |d(s)|∞ ≤ R−3e−c0s/R. Consequently,∫ t

0

|d(s)|∞ ds ≤ cR−2 ≤ ln 2,

We have arrived at

(355)1

2R≤ ρ(t), ρ(t) ≤ 2R.

Next we obtain higher order bounds on ρ with the help of the e-quantity. Note that the right hand side ofthe e-equation (233) is bounded by

|(∇ · u)2 − Tr(∇u)2| ≤ c[u]21 . R−6e−c0t/R.

So, from (233) we obtaind

dt|e|∞ ≤ R−3e−c0t/R|e|∞ +R−6e−c0t/R.

By the Gronwall inequality, and using that |e0|∞ < R, we conclude

(356) |e(t)|∞ ≤ 2R, t < t∗.

By a similar computation for ∇e we obtain

d

dt[e]1 . [u]1[e]1 + [u]2|e|∞ + c[u]1[u]2,

and using (354),d

dt[e]1 . R

−3e−c0t/R[e]1 + 2R−2e−c0t/R + cR−3e−c0t/R.

Since initially [e0]1 ≤ [u0]2 + [ρ0]3 < 2R, again by the Gronwall inequality,

[e]1 ≤ 4R.

This implies that [Λαρ]1 < 5R. So, if α 6= 1, this translates directly into the Holder norm and we obtain

(357) [ρ]1+α ≤ c0R,

110 ROMAN SHVYDKOY

while for α = 1, however it implies bounds in other border-line classes, and as a consequence,

(358) [ρ]2−γ ≤ c0R.

Step 3: higher order non-local maximum principle. Here we adapt the argument of Lemma 9.9 toobtain the non-local maximum principle for higher order finite differences. As before we denote

(359) Dαf(x) =

∫Rn|f(x+ z)− f(x)|2 dz

|z|n+α.

Lemma 10.9. There is an absolute constant c0 > 0 such that

(360) Dα[δ3hf ](x) ≥ c0

|δ3hf(x)|2+α

[f ]α2 |h|3α.

Proof. Fix a smooth cut-off function χ, and r > 0 to be specified later. We obtain

Dα[δ3hf ](x) ≥

∫Rn|δzδ3

hf(x)|2 1− χ(z/r)

|z|n+αdz

≥∫Rn

(|δ3hf(x)|2 − 2δ3

hf(x)δ3hf(x+ z))

1− χ(z/r)

|z|n+αdz

≥ |δ3hf(x)|2 1

rα− 2δ3

hf(x)

∫Rnδ3hf(x+ z)

1− χ(z/r)

|z|n+αdz.

Note the Taylor residue formula

δ3hf(x+ z) =

∫ 1

0

∫ 1

0

∫ 1

0

∇3zf(x+ z + (θ1 + θ2 + θ3)h)(h, h, h) dθ1 dθ2 dθ3.

Integrating by parts in z, and using the bound∣∣∣∣∇z (1− χ(z/r)

|z|n+α

)∣∣∣∣ ≤ c

|z|n+α+1χ|z|>r,

we obtain ∣∣∣∣∫Rnδ3hf(x+ z)

1− χ(z/r)

|z|n+αdz

∣∣∣∣ ≤ C[f ]2|h|3rα+1

.

Continuing with main estimate we obtain

Dα[δ3hf ](x) ≥ |δ3

hf(x)|2 1

rα− C[f ]2|δ3

hf(x)| |h|3

rα+1.

Choosing r ∼ [f ]2|δ3hf(x)||h|3|δ3hf(x)|2 produces (360).

Step 4: Main estimates. We are now in a position to use the velocity equation to make estimates on thederivative of [u]2+γ . Let us fix a pair (x, h) ∈ Tn which maximizes (349), and we have at time t∗

[u]2+γ =|δ3hu(x)||h|2+γ

= R.

Consequently, at this time, using (360)

(361)1

|h|4+2γDαδ

3hu(x) ≥ R8+α

|h|α(1−γ).

Let us now write the equation for the third finite difference:

(362) ∂tδ3hu + δ3

h(u · ∇u) =

∫Rnδ3h[ρ(x+ z)δzu(x)]

dz

|z|n+α.

Let us denote the transport and alignment terms by

B = δ3h(u · ∇u),

I =

∫Rnδ3h[ρ(x+ z)δzu(x)]

dz

|z|n+α.

(363)


Let use the test-function δ3hu(x)/|h|4+2γ and evaluate at the maximizing pair at which point we also have

δ3hu(x)

|h|4+2γ=

[u]2+γ

|h|2+γ.

So we obtain from the equation,

∂t[u]22+γ +[u]2+γ

|h|2+γB =

δ3hu(x)

|h|4+2γI.

Let us first estimate the transport term. By the product formula

δ3h(fg) = δ3

hfτ3hg + 3δ2hfδhτ2hg + 3δhfδ

2hτhg + fδ3

hg.

Thus,

B = δ3hu · τ3h∇u + 3δ2

hu · δhτ2h∇u + 3δhu · δ2hτh∇u + u · ∇δ3

hu.

Note that the last term vanishes due to criticality. Consequently,

1

|h|2+γ|B| ≤ [u]2+γ [u]1 + 3|h|1−γ [u]22 + 3[u]1[u]2+γ . [u]2+γ [u]1 + |h|1−γ [u]22.

Multiplying by another [u]2+γ ≤ R and using (354) we obtain

(364)[u]2+γ

|h|2+γ|B| . R−1 +R−5 < 1.

We now turn to the dissipation term. The integrand is given by δ3h[τzρ δzu]. So, we expand by the product

rule and using commutativity δhδz = δzδh:

δ3h[τzρ δzu] = δ3

hτzρ τ3hδzu + 3δ2hτzρ τ2hδhδzu + 3δhτzρ τhδ

2hδzu + τzρ δzδ

3hu.(365)

Multiplying by δ3hu the last term provides necessary dissipation:

τzρ δzδ3hu δ

3hu ≤ −

1

2ρ |δzδ3

hu|2.

Dividing by |h|4+2γ and using (361) we obtain the lower bound

(366)1

2|h|4+2γρDαδ

3hu(x) ≥ R7

|h|α(1−γ).

In particular we can see that the entire transport term estimated in (364) is absorbed by the dissipation atthe time t∗. As a result we obtain the equation

∂t[u]22+γ ≤ −R7

|h|α(1−γ)+δ3hu(x)

|h|4+2γJ,

where J contains all the remaining three terms of I:

J =

∫Rn

[δ3hτzρ τ3hδzu + 3δ2

hτzρ τ2hδhδzu + 3δhτzρ τhδ2hδzu]

dz

|z|n+α= J1 + 3J2 + 3J3.

For the remainder of the proof we provide estimates for each of the Ji terms with the common goal to obtainthe bound

(367)1

|h|2+γ|Ji| .

|h|ε|h|α(1−γ)

,

for some ε > 0 and γ is sufficiently small. If this is achieved, then the dissipation absorbs all these remainingJ-terms and we conclude that

∂t[u(t∗)]22+γ < 0,

which would finish the proof.So, let us begin with J1. Symmetrizing in z we obtain

(368) J1 =1

2

∫Rn

[δ3h(τzρ− τ−zρ) τ3hδzu + δ3

hτzρ τ3h(δzu+ δ−zu)]dz

|z|n+α.

For the first summand we use

|δ3h(τzρ− τ−zρ) τ3hδzu| ≤ |h|α−γ min|z|2, |h|.

112 ROMAN SHVYDKOY

For the second summand,

|δ3hτzρ τ3h(δzu + δ−zu)| ≤ |h|1+α−γ min|z|2, 1.

Plugging into (368) and integrating we obtain the desired (367), but the computation extends only up toα > 1

2 . The problem is that the density receives all the δh’s and not fully utilizes them. At the same time,u can no longer directly contribute powers of h. So, we switch one h-difference back onto u. So, let us fix0 < α ≤ 1

2 . We start from the original formula

J1 =

∫Rnδ3hτzρ(x) τ3hδzu(x)

dz

|z|n+α.

Over the domain |z| < 10|h| we estimate using a cut-off function χ as before,∫Rn|δ3hτzρ(x) τ3hδzu(x)|χ

(z

10|h|

)dz

|z|n+α≤∫|z|<10|h|

|h|1+α dz

|z|n+α−1. |h|2.

This culminates into (367). For the remaining part, denote for clarity f = δ2hρ. So, δ3

hτzρ(x) = f(x + h +z)− f(x+ z). Let us write∫

Rn(f(x+ h+ z)− f(x+ z)) τ3hδzu(x)

(1− χ( z10|h| )) dz

|z|n+α

=

∫Rnf(x+ z)

(τ3hδz−hu(x)

(1− χ( z−h10|h| ))

|z − h|n+α− τ3hδzu(x)

(1− χ( z10|h| ))

|z|n+α

)dz

=

∫Rnf(x+ z)τ3h(δz−hu(x)− δzu(x))

(1− χ( z−h10|h| ))

|z − h|n+αdz

−∫Rnf(x+ z)τ3hδzu(x)

((1− χ( z−h10|h| ))

|z − h|n+α−

(1− χ( z10|h| ))

|z|n+α

)dz

All the integrals are supported on |z| > 9|h|, where |z − h| ∼ |z|. Estimating the first one we use

|δz−hu(x)− δzu(x)| = |u(x+ z − h)− u(x+ z)| ≤ |h||f(x+ z)| ≤ |h|1+α.

Consequently,∣∣∣∣∣∫Rnf(x+ z)τ3h(δz−hu(x)− δzu(x))

(1− ψ( z−h10|h| ))

|z − h|n+αdz

∣∣∣∣∣ ≤ |h|2+α

∫|z|≥|h|

dz

|z|n+α≤ |h|2,

which implies (367). For the second integral we use∣∣∣∣∣ (1− ψ( z−h10|h| ))

|z − h|n+α−

(1− ψ( z10|h| ))

|z|n+α

∣∣∣∣∣ ≤ |h||z − θh|n+α+1

1|z|>9|h| .|h|

|z|n+α+11|z|>9|h|,

and

|f(x+ z)τ3hδzu(x)| ≤ |h|1+α|z|.Integration reproduces the same bound as for the first part.

Next, J2. For α < 1, let us use (354) and (357) to deduce

|δ2hτzρ| ≤ [ρ]1+α|h|1+α . R|h|1+α

|τ2hδhδzu| ≤ [u]2|h|min|z|, 1 . R−1|h|min|z|, 1.The singularity is now removed and we get

1

|h|2+γ|J2| . |h|α−γ ,

which implies (367) for sufficiently small γ. For α ≥ 1 we first symmetrize

J2 =1

2

∫Rn

[δ2h(τz − τ−z)ρ τ2hδhδzu + δ2

hτzρ τ2hδh(δz + δ−z)u]dz

|z|n+α.


Here for the first summand we use (358):

|δ2h(τz − τ−z)ρ| ≤ Rmin|h|1+α−γ , |h|α−γ |z||τ2hδhδzu| ≤ R−1|h|min|z|, 1.

With this at hand we proceed

1

|h|2+γ

∫Rn|δ2h(τz − τ−z)ρ τ2hδhδzu|

dz

|z|n+α≤ |h|

1+α−γ

|h|2+γ≤ |h|

α−2γ−1+α(1−γ)

|h|α(1−γ)≤ |h|ε|h|α(1−γ)

,

since clearly, α − 2γ − 1 + α(1− γ) > 0 for small g. In the second summand, using that (δz + δ−z)u is thesecond order finite difference,

|δ2hτzρ τ2hδh(δz + δ−z)u| ≤ |h|2−γ min|z|2, 1

we obtain1

|h|2+γ

∫Rn|δ2hτzρ τ2hδh(δz + δ−z)u|

dz

|z|n+α≤ |h|

2−γ

|h|2+γ≤ hα(1−γ)−2γ

|h|α(1−γ).

This finishes the bound on J2.Finally, for J3 we proceed similarly. For α < 1, we use

|δhτzρ τhδ2hδzu| ≤ |h|2 min|z|, 1.

Hence,

1

|h|2+γ|J3| .

|h|2|h|2+γ

≤ hα(1−γ)−γ

|h|α(1−γ)≤ hε

|h|α(1−γ).

For α ≥ 1, we again symmetrize first

J3 =1

2

∫Rn

[δh(τz − τ−z)ρ τhδ2hδzu + δhτzρ τhδ

2h(δz + δ−z)u]

dz

|z|n+α,

and using

|δh(τz − τ−z)ρ τhδ2hδzu| ≤ min|h|3+α−γ , |h|1+α|z|2,

|δhτzρ τhδ2h(δz + δ−z)u| ≤ min|h|3, |h||z|2,

integration implies (367).We have established that ∂t[u(t∗)]22+γ < 0 at the critical time. This means that such time t∗ does not

exist and which finishes the proof of existence part.

Step 5: Flocking and Stability. As we noted in the beginning, exponential decay of [u]22 implies uniformcontrol over |∇ρ|∞.

Arguing as in the proof of Theorem 9.1 we can slightly improve the space in which strong flocking occurs.This is due to (357) - (358) bounds, which imply that ρ ∈ W 1+α−γ,∞ by compactness. Using again (358)and by interpolation we have convergence in the W 1,∞-metric as well:

[ρ(·, t)− ρ(·, t)]1 < C2e−δt.

As far stability is concerned, the computation above shows that in fact the limiting profile r differs littlefrom initial density r0 under the conditions of Theorem 10.8. Indeed, setting R such that ε = 1/RN (hereε > 0 is small), we obtain via (354),

|∂tr|∞ ≤ CR−2e−c0t/R.

Hence, |r − r0|∞ ≤ Cc0R

= εθ. Since |r0 − ρ0|∞ < ε, this finishes the result.

10.5. Notes and References. The results of Section 10.1 and 10.2 are taken from an upcoming [63], seealso [64] for an extension to singular models. The spectral dynamics approach was initially proposed inTadmor and Tan [94], and reached its present form in He and Tadmor [51]. Work [94] also examines variousthreshold conditions of regularity in 2D settings. It remains open whether there is an umbrella family ofsolutions that would incorporate both unidirectional and small spectral gap classes.

Results of Section 10.4 were proved in [87]. An alternative approach appeared in Danchin et al [35] withsmallness condition formulated in Besov spaces with positive smoothness. Due to lack of a good continuationcriterion, it is not clear whether similar results hold for topological models.

114 ROMAN SHVYDKOY

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University of Illinois at Chicago, 60607E-mail address: [email protected]

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