hydrodynamic limit of the kinetic cucker-smale flocking modelmellet//publi/flocking_hydro.pdf ·...

HYDRODYNAMIC LIMIT OF THE KINETICCUCKER-SMALE FLOCKING MODEL

TRYGVE K. KARPER, ANTOINE MELLET, AND KONSTANTINA TRIVISA

Abstract. The hydrodynamic limit of a kinetic Cucker-Smale flocking model

is investigated. The starting point is the model considered in [17], which in ad-

dition to free-transport of individuals and a standard Cucker-Smale alignmentoperator, includes Brownian noise and strong local alignment. The latter was

derived in [18] as the singular limit of an alignment operator first introduced

by Motsch and Tadmor in [25]. The objective of this work is the rigorous in-vestigation of the singular limit corresponding to strong noise and strong local

alignment. The proof relies on a relative entropy method. The asymptoticdynamics is described by an Euler-type flocking system.

Contents

1. Introduction 12. Existence theory 53. Main result 74. Proof of Theorem 3.1 8Appendix A. Local well-posedness of the Euler-flocking system 19Appendix B. Entropy inequality 22References 25

1. Introduction

In nature, a variety of phenomena are observed daily in which the interactionsamong individuals in a population gives rise to large scale coherent structures. Themotivation behind this work is the self-organized movement of agents observedwithin a flock of birds, school of fish, or swarm of insects [1, 3, 20]. Mathematicalmodeling of such phenomena is currently at the center of intense scientific researchand many mathematical models have been proposed to capture parts of the observedbehavior. Some popular examples are the models presented in [6, 9, 7, 23, 25, 27].The first models to be introduced were particle systems describing the behaviorof each individual within a population following a simple set of rules. From thesemodels, kinetic and hydrodynamic models have been derived. These models areless complex than particle systems and are typically better suited to the study offlocking phenomena when the number of individuals in the flock is very large. To

Date: May 1, 2013.

1991 Mathematics Subject Classification. Primary:35Q84; Secondary:35D30.Key words and phrases. flocking, kinetic equations, hydrodynamic limit, relative entropy,

Cucker-Smale, self-organized dynamics.The work of T.K was supported by the Research Council of Norway through the project 205738.The work of A.M was supported by the National Science Foundation under the Grant DMS-

0901340.The work of K.T. was supported by the National Science Foundation under the Grants DMS-

1109397 and DMS-1211638.

1

2 KARPER, MELLET, AND TRIVISA

date, the majority of rigorous studies on flocking models have concerned particlesystems and the corresponding kinetic equations.

In this paper, our contribution will be the rigorous derivation of an Euler typehydrodynamic model for flocking. Our starting point is the following kinetic Cucker-Smale equation [7, 16, 17] on Rd × Rd × (0, T )

ft + v · ∇xf + divv (fL[f ]) = σ∆vf + β divv(f(v − u)). (1.1) eq:eq1

Here, f := f(t, x, v) is the scalar density of individuals, d ≥ 1 is the spatial di-mension, and β, σ ≥ 0 are some constants. The alignment operator L is the usualCucker-Smale (CS) operator, which has the form

L[f ] =∫

Rd

∫RdK(x, y)f(y, w)(w − v) dw dy, (1.2) eq:CS

with K being a smooth symmetric kernel. The last term in (1.1) describes stronglocal alignment interactions, where u denotes the average local velocity, defined by

u(t, x) =

∫Rd fv dv∫Rd f dv

.

This model was first introduced in [17], in which weak solutions were proven toexist for all finite time. In [18], model (1.1) was rigorously derived as the singularlimit of a kinetic model proposed by Motsch and Tadmor in [25]. Motivation forthis model is briefly presented in Section 1.1 below.

Since the kinetic equation (1.1) is posed in 2d+1 dimensions, obtaining a numer-ical solution of (1.1) is very costly. In fact, the most feasible approach seems to beMonte-Carlo methods using solutions of the underlying particle model with a largenumber of particles and realizations. Consequently, it is of great interest to deter-mine parameter regimes where the model may be reduced in complexity. The goalof this paper is to study the singular limit of (1.1) corresponding to strong noiseand strong local alignment, that is σ, β → ∞. More precisely, we are concernedwith the limit ε→ 0 in the following equation:

fεt + v · ∇xfε + divv (fεL[fε]) =1ε

∆vfε +

1ε

divv(fε(v − uε)). (1.3) eq:eq2

This scaling can alternatively be obtained from (1.1) by the change of variables

x = εx, t = εt,

and assuming that K(x, y) = εK(x− y).In this paper, we shall establish with rigorous arguments that under suitable

assumptions on the initial data, we have

fε → %(t, x)e|v−u(t,x)|2

2 ,

where % and u solve the following Euler-Flocking system

%t + divx(%u) = 0, (1.4) eq:cont

(%u)t + divx(%u⊗ u) +∇x% =∫

RdK(x, y)%(x)%(y)[u(y)− u(x)] dy. (1.5) eq:moment

The result will be precisely stated in Theorem 3.1 with the proof coming up inSection 4. The proof is established via a relative entropy argument which will alsoprovide a rate of convergence. This relative entropy method relies on the “weak-strong” uniqueness principle established by Dafermos for systems of conservationlaws admitting convex entropy functional [10] (see also [11]). It has been successfullyused to study hydrodynamic limits of particle systems [15, 19, 22, 24, 28]. As often,the result will be somewhat restricted as we need the existence of smooth solutionsto (1.4) which we only know locally in time.

HYDRODYNAMIC LIMIT OF THE KINETIC CUCKER-SMALE FLOCKING MODEL 3

It should be mentioned that this is not the first paper to investigate hydro-dynamic descriptions of flocking behavior. In particular, we refer the reader to[4, 12, 16, 25], where kinematic closure relations are either assumed or formallymotivated and hydrodynamic descriptions of the form (1.4) - (1.5) are derived.Other approaches have also been considered, such as those in [14, 26].

What distinguishes the present article from earlier works on related models isthe fact that it provides a rigorous proof of the limit and a rate of convergence, inthe regime corresponding to strong noise and strong local alignment.

1.1. Derivation of the model. In this section, we briefly recall the origin of thekinetic model (1.1), which is presented with further details in [17, 18]. Our startingpoint is the pioneering model introduced by Cucker and Smale [7]: Consider Nindividuals (birds, fish) each totally described by a position xi(t) and a velocityvi(t). The Cucker-Smale (CS) model [7, 8] corresponds to the evolution equations

xi = vi, vi = − 1N

∑j 6=i

K0(xi, xj)(vj − vi), (1.6) eq:particle

which states that each individual align its velocity with a local average velocity.By direct calculation, one sees that the empirical distribution function

f(t, x, v) =1N

∑i

δ(x− xi(t))δ(v − vi(t)),

is a distributional solution to the kinetic equation

ft + v · ∇xf + divv (fL[f ]) = 0, (1.7) eq:eq-1

where L is given by

L[f ] =∫

Rd

∫RdK0(x, y)f(y, w)(w − v) dw dy. (1.8) eq:CSop

In [25], Motsch and Tadmor identify a significant drawback of the CS model(1.7) which is due to the normalization factor 1

N in (1.6). Specifically, if a smallgroup of individuals are located far away from a much larger group of individuals,the internal dynamics in the small group is almost halted since the total numberof individuals is large. To remedy this, they propose a new model (MT), in which(1.8) is replaced with a normalized non-symmetric alignment operator of the form

L[f ] =

∫Rd∫Rdφ(x− y)f(y, w)(w − v) dw dy∫

Rd∫Rdφ(x− y)f(y, w) dw dy

(1.9) eq:MT

for some kernel φ.It is reasonable to combine the CS model with the MT model, letting the CS

flocking term (1.8) dominate the long-range interaction and the MT term (1.9)dominate short-range interactions. This will correct the aforementioned deficiencyof the kinetic CS model. However, the large-range interactions is still close to thatof the CS model. Since the MT alignment term governs short-range interactions atthe particle level, it makes sense to consider the singular limit where φ convergesto a Dirac distribution at the mesoscopic level. The MT correction then convergesto a local alignment term given by:

L[f ] =j − ρvρ

= u− v.

where

ρ(x, t) =∫

Rdf(x, v, t) dv, j(x, t) =

∫Rdvf(x, v, t) dv


and u(x, t) is defined by the relation

u =

∫Rd vf dv∫Rd f dv

. (1.10) ?eq:uu?

This leads to the following equation:

ft + v · ∇xf + divv (fL[f ]) + β divv(f(u− v)) = 0,

with L given by (1.8). By adding Brownian noise, we arrive at (1.1). Note thatthe addition of this noise term is mainly for technical reasons. It is not necessaryto the formal derivation of a hydrodynamic limit (it is responsible for the pressureterm ∇xρ in (1.5)). But it plays an important role in our proof by including theterm f log f in the entropy.

1.2. Formal derivation of (1.4) - (1.5). For the convenience of the reader, wenow present the formal arguments leading to (1.4) - (1.5). First, we note thatwhen ε → 0, the right-hand side in (1.3) should converge to zero and so the limitf = limε→0 f

ε should satisfy

∆vf + divv(f(v − u)) = 0

and thus be of the form

f(t, x, v) = %(t, x)e−|v−u(t,x)|2

2 .

Hence, it seems plausible that the evolution of f (in the limit) can be governed byequations for the macroscopic quantities % and u alone.

To derive equations for % and u, we first integrate (1.3) with respect to v to get

%εt + div(%εuε) = 0. (1.11) eq:cont0

Passing to the limit (formally) yields the continuity equation (1.4).Next, we multiply (1.3) by v and integrate with respect to v to obtain

(%εuε)t + divx

(∫Rd

(v ⊗ v)fε dv)

−∫

RdK(x, y)%ε(x)%ε(y)(uε(x)− uε(y)) dy = 0.

(1.12) eq:mom

Passing to the limit in (1.12), we get:

(%u)t + divx

(%

∫Rd

(v ⊗ v)e−|v−u|2

2 dv

)−∫

RdK(x, y)%(x)%(y)(u(x)− u(y)) dy = 0.

(1.13) eq:jalla

By adding and subtracting u, we discover that∫Rd

(v ⊗ v)e−|v−u|2

2 dv =∫

Rd(u⊗ u)e−

|v−u|22 + (v − u)⊗ (v − u)e−

|v−u|22 dv

= u⊗ u+ I.

Inserting this expression in (1.13) gives (1.5).

Organization of the paper: The rest of this paper is organized as follows: InSection 2, we recall some existence results for the kinetic flocking model (1.1) (theseresults were proved in [17]) and for the Euler-flocking model (1.4)-(1.5) (a proof ofthis result is provided in Appendix A). In Section 3 we present our main result,which establishes the convergence of weak solutions of the kinetic equation (1.3) tothe unique strong solution of the Euler-flocking system (1.4)-(1.5). The proof ofthe main theorem is then developed in Section 4.


2. Existence theory〈S2〉 In this section we present some existence results that will be of use in the se-

quel. More precisely, the proof of our main result (convergence of (1.3) to (1.4)-(1.5)) makes use of relative entropy arguments which require the existence of weaksolutions of (1.3) satisfying an appropriate entropy inequality, and the existenceof strong solutions to the Euler-Flocking system (1.4)-(1.5) satisfying an entropyequality. Note that the latter result will be obtained only for short time.

Since entropies play a crucial role throughout the paper, we first need to presentthe entropy equalities and inequalities satisfied by smooth solutions of (1.3) and(1.4)-(1.5).

2.1. Entropy inequalities. Solutions of (1.1) satisfy an important entropy equal-ity, which was derived in [17] and can be found in the appendix: We define theentropy

F(f) =∫

R2df log f + f

|v|2

2dv dx (2.1) entropy

and the dissipations

D1(f) =∫

R2d

1f|∇vf − f(u− v)|2 dv dx,

D2(f) =12

∫Rd

∫Rd

∫Rd

∫RdK(x, y)f(x, v)f(y, w) |v − w|2 dw dy dv dx

(2.2) dissipation

(the latter is the dissipation associated with the CS operator L[·]). We then have:

〈prop:main〉Proposition 2.1. Assume that L is the alignment operator given by (1.2) with Ksymmetric and bounded. If f is a smooth solution of (1.3), then f satisfies

∂tF(f) +1εD1(f) +D2(f)

= d

∫Rd

∫Rd

∫Rd

∫RdK(x, y)f(x, v)f(y, w) dw dy dv dx, (2.3) eq:entropy1

with F(·), D1(·), D2(·) given by (2.1) and (2.2).Furthermore, there exists C, depending only on T , ‖K‖∞ and

∫f0(x, v) dx dv,

such that

∂tF(f) +12εD1(f)

+12

∫Rd

∫RdK(x, y)%(x)%(y) |u(x)− u(y)|2 dy dx ≤ C(T )ε

(2.4) eq:entropy1.2

for all t ≤ T .

The first inequality (2.3) shows that the nonlocal alignment term is responsiblefor some creation of entropy. The second inequality (2.4) shows that this term canbe controlled by D1(f) and the entropy itself. This last inequality will play a keyrole in this paper.

The Euler system of equations (1.4)-(1.5) also satisfies a classical entropy equal-ity. More precisely, if we define

E(%, u) =∫

Rd%u2

2dx+

∫Rd% log % dx,

then any smooth solution of (1.4)-(1.5) satisfies

∂tE(%, u) +12

∫Rd

∫RdK(x, y)%(x)%(y) |u(x)− u(y)|2 dy dx = 0.


Note that the entropy E and F are related to each other by the relation

F(

%

(2π)d/2e−|v−u|2

2

)= E(%, u).

Furthermore, we have the following classical minimization principle (consequenceof Jensen inequality):

E(%, u) ≤ F(f), if % =∫f dv, %u =

∫vf dv. (2.5) eq:min

This relation will be important in the upcoming analysis when considering therelative entropy of solutions to the kinetic equation (1.3) and solutions to (1.4) -(1.5).

2.2. Global weak solutions of the kinetic equation. The existence of a weaksolution for (1.1) is far from trivial because of the singularity in the definition of u.We will say that a function f satisfying

f ∈ C(0, T ;L1(R2d)) ∩ L∞((0, T )× R2d), (|v|2 + |x|2)f ∈ L∞(0,∞;L1(R2d)),

is a weak solution of (1.1) if the following holds:∫R2d+1

−fψt − vf∇xψ − fL[f ]∇vψ dvdxdt

+∫

R2d+1σ∇vf∇vψ − βf(u− v)∇vψ dvdxdt

=∫

R2df0ψ(0, ·) dvdx,

(2.6) eq:weak

for any ψ ∈ C∞c ([0, T )× R2d), where u is such that j :=∫fv dv = %u.

Remark 2.2. Note that the definition of u is ambiguous if % vanishes (vacuum). Weresolve this by defining u pointwise as follows

u(x, t) =

j(x, t)%(x, t)

if %(x, t) 6= 0

0 if %(x, t) = 0(2.7) eq:uu1

This gives a consistent definition of u as can be seen from the bound

j ≤(∫|v|2f(x, v, t) dv

)1/2

%1/2,

yielding j = 0 whenever % = 0 and so (2.7) implies j = %u.Note also that u does not belong to any Lp space. However, we have∫

R2d|uf |2 dx dv ≤ ‖f‖L∞(R2d)

∫R2d|v|2f(x, v, t) dv dx,

so the term uf in the weak formulation (2.6) makes sense as a function in L2.

To state the existence result we shall use

F+(f) =∫

R2df | log f |+ f

|v|2

2dvdx.

The existence part of the following theorem was obtained as the main result in [17].

〈thm:kinetic〉Theorem 2.3. Assume that L is the alignment operator (CS) given by (1.2) withK non-negative, symmetric and bounded. Assume furthermore that f0 satisfies

f0 ∈ L∞(R2d) ∩ L1(R2d), and (|v|2 + |x|2)f0 ∈ L1(R2d).


Then, for all ε > 0, there exist a weak solution fε of (1.3) satisfying

F+(fε(t)) +∫ t

0

1εD1(fε) +D2(fε) ds ≤ C(T ) (2.8) eq:entropyfinal

where the constant C depends only on T , ‖K‖∞ and∫f0(x, v) dx dv.

Furthermore,

F(fε(t)) +12ε

∫ t

0

D1(fε)ds+12

∫ t

0

∫R2d

K(x, y)%ε(x)%ε(y) |uε(x)− uε(y)|2 dydxds

≤ F(f0) + C(T )ε (2.9) eq:entropyfinal2

for all t > 0.

The entropy inequality (2.8) is proved in the appendix in Lemma B.2, while (2.9)is a consequence of (2.4).

2.3. Existence of solutions to the Euler-Flocking system. In the upcominganalysis, we will need that (1.4) - (1.5) admits a strong solution, at least for shorttime. This is the object of the next theorem (which we state in the case d = 3):

〈thm:Euler〉Theorem 2.4. Let (ρ0, u0) ∈ Hs(R3) with s > 5/2 and ρ0(x) > 0 in R3. Then,there exist T ∗ > 0 and functions (%, u) ∈ C([0, T ∗];Hs(R3))∩C1((0, T ∗);Hs−1(R3)),ρ(x, t) > 0, such that (%, u) is the unique strong solution of (1.4) - (1.5) fort ∈ (0, T ∗). Moreover, (%, u) satisfies the equality

∂tE(%, u) +12

∫Rd

∫RdK(x, y)%(x)%(y) |u(x)− u(y)|2 dydx = 0. (2.10) eq:entropyEuler

Since the proof of this theorem is rather long and independent of the rest of thepaper, we postpone it to the appendix.

Note in particular that the condition s > 5/2 implies that the solution satisfies

u ∈ L∞([0, T ∗];W 1,∞(R3)). (2.11) eq:bdu

Furthermore, dividing the momentum equation by ρ, we also get:

∇x log ρ ∈ L∞([0, T ∗]× R3). (2.12) eq:bdlog

These two estimates is all the regularity we will need in our main theorem below.

3. Main result〈S3〉 With the existence results of the previous section, we are ready to state our main

result concerning the convergence of weak solutions of (1.3) to the strong solution(%, u) of the Euler-flocking system (1.5)-(1.4) as ε→ 0.

〈thm:main〉Theorem 3.1. Assume that:(1) f0 is of the form

f0 =%0(x)

(2πd/2)e−|u0(x)−v|2

2 ,

with

f0 ∈ L∞(R2d) ∩ L1(R2d), and (|v|2 + |x|2)f0 ∈ L1(R2d).

(2) fε is a weak solution of (1.3) satisfying the entropy inequality (2.9) andwith initial condition fε(0, ·) = f0(·).


(3) T ∗ > 0 is the maximal time for which there exists a strong solution (%, u)to the Euler system of equations (1.4) - (1.5), with %0 =

∫Rd f0 dv and

%0u0 =∫

Rd f0v dv and satisfying (2.11) and (2.12) (Theorem 2.4 gives inparticular T ∗ > 0 if ρ0 and u0 are regular enough).

There exists a constant C > 0 depending on

F(f0), ‖K‖L∞ , T ∗, ‖u‖L∞(0,T∗;W 1,∞(Rd)), and ||∇ log ρ||L∞((0,T∗)×Rd),

such that∫ T∗

0

∫Rd

%ε

2|uε − u|2 +

∫ %ε

%

%ε − zz

dz dxdt (3.1) eq:main

+12

∫ T∗

0

∫Rd

∫RdK(x, y)%ε(x)%ε(y) [(uε(x)− u(x))− (uε(y)− u(y))]2 dxdydt

≤ C√ε,

where%ε =

∫Rdfε dv, %εuε =

∫Rdvfε dv.

Moreover, any sequence of functions satisfying (3.1) satisfies:

fεε→0−→ %e−

|v−u|22 a.e and L1

loc(0, T∗;L1(Rd × Rd)),

%εε→0−→ % a.e and L1

loc(0, T∗;L1(Rd)),

%εuεε→0−→ %u a.e and L1

loc(0, T∗;L1(Rd)),

%ε|uε|2 ε→0−→ %u2 a.e and L1loc(0, T

∗;L1(Rd)).

Remark 3.2. We note that instead of assuming that the initial data is well prepared(condition (1) in the theorem above), we could assume that fε is a weak solutionof (1.3) with initial data fε0 such that∫ T∗

0

∫Rd

%ε02|uε0 − u0|2 +

∫ %ε0

%0

%ε0 − zz

dz dx dt ≤ C√ε.

This theorem will be a direct consequence of Proposition 4.1 below, the proof ofwhich is the object of Section 4.

4. Proof of Theorem 3.1〈sec:proof〉 To simplify notation in the proof of Theorem 3.1 we now express the Euler-

Flocking system in terms of the conservative quantities. In the present context, theconservative quantities are the density % and the momentum P = %u. If we denote

U =(%P

),

we can rewrite the system (1.4)-(1.5) as

Ut + divxA(U) = F (U). (4.1) def:vec

The flux and source term are then given by

A(U) =(

P 0P⊗P% %

), F (U) =

(0

%P − %P

),

where we have introduced the notation g =∫

Rd K(x, y)g(y) dy. The entropy Ecorresponding to (4.1) reads

E(U) =P 2

2%+ % log %,


and the relative entropy is the quantity

E(V |U) = E(V )− E(U)− dE(U)(V − U),

where d stands for the derivation with respect to the variables (%, P ).

For the system (1.4)-(1.5), a simple computation yields

−dE(U)(V − U) = −

(− P 2

2%2 + log %+ 1P%

)(q − %Q− P

)=q|u|2

2− %|u|2

2+ (%− q)(log %+ 1) + %u2 − quv.

Conseqently, the relative entropy can alternatively be writtenE(V |U) = E(V )− E(U)− dE(U)(V − U)

= q|v|2

2− % |u|

2

2+ q log q − % log %

+q|u|2

2− %|u|2

2+ (%− q)(log %+ 1) + %u2 − quv

= q|v − u|2

2+ p(q|%),

(4.2) eq:relativecontrol

where we have introduced the relative pressure

p(q|%) = q log q − % log %+ (%− q)(log %+ 1) =∫ q

%

q − zz

dz.

Note that the relative pressure controls the L2 norm of the difference

p(q|%) ≥ 12

min

1q(x)

,1

%(x)

(q(x)− %(x))2. (4.3) eq:relativepressure

With the newly introduced notation, Theorem 3.1 can be recast as a directconsequence of the following proposition:

〈prop:relative〉Proposition 4.1. Under the assumptions of Theorem 3.1, let

U =(%%u

)denote the strong solution to the Euler system of equations (1.4)-(1.5) and let

U ε =(%ε

%εuε

), %ε =

∫Rdf ε dv, %εuε =

∫Rdf εv dv,

be the macroscopic quantities corresponding to the weak solution of the kinetic equa-tion (1.3).

The following inequality holds:∫RdE(U ε|U)(t) dx

+12

∫ t

0

∫Rd

∫RdK(x, y)%ε(x)%ε(y) [(uε(x)− u(x))− (uε(y)− u(y))]2 dx dy ds

≤ C∫ t

0

∫RdE(Uε|U) dx ds+ C

√ε.

(4.4) eq:final

The proof of this proposition relies on several auxiliary results which will bestated and proved throughout this section. At the end of the section, in Section 4.6we close the arguments and conclude the proof. However, before we continue, letus first convince the reader that Proposition 4.1 actually yields Theorem 3.1.


Proof of Theorem 3.1. Let us for the moment take Proposition 4.1 for granted.Then, the main inequality (3.1) follows from (4.5) and Gronwall’s lemma.

We now need to show that (3.1) implies the stated convergence. First, we notethat the entropy estimate (2.8) implies that fε is bounded in L logL and thusconverges weakly to some f .

Next, in view of (4.2) and (4.3), the main inequality (3.1) yields∫ T∗

0

∫Rd

min

1ρε,

1ρ

|ρε − ρ|2 dx dt −→ 0

and ∫ T∗

0

∫Rdρε|uε − u|2 dx dt −→ 0.

In particular, we get:∫Rd|ρε − ρ| dx =

∫Rd

min

1ρε,

1ρ

1/2

maxρε, ρ1/2|ρε − ρ| dx

≤(∫

Rdmin

1ρε,

1ρ

|ρε − ρ|2 dx

)1/2(∫Rd

maxρε, ρdx)1/2

≤(∫

Rdmin

1ρε,

1ρ

|ρε − ρ|2 dx

)1/2

(2M)1/2.

Hence, we can conclude that

%εε→0−→ % a.e and L1

loc(0, T∗;L1(Rd)),

Similary, we see that∫Rd|%εuε − %u| dx ≤

∫Rd|%ε(uε − u)|+ |(%ε − %)u| dx

≤M 12

(∫Rd%ε|uε − u|2 dx

)+(∫

Rdmin

1ρε,

1ρ

|ρε − ρ|2 dx

)1/2(∫Rd

maxρε, ρu2 dx

)1/2

.

Consequently, also

%εuεε→0−→ %u a.e and L1

loc(0, T∗;L1(Rd)).

Moreover, by writing

ρεuε2 − ρu2 = ρε(uε − u)2 + 2u(ρεuε − ρu) + u2(ρ− ρε)

we easily deduce

%ε|uε|2 ε→0−→ %u2 a.e and L1loc(0, T

∗;L1(Rd)).

At this stage, it only remains to prove that f has the stated maxwellian form.For this purpose, we first send ε → 0 in the entropy inequality (2.8) and use theconvergence of %εuε and %ε|uε|2 to obtain

limε→0

∫Rdfεv2

2+ fε log fε dx+

12

∫ t

0

∫RdK(x, y)%(x)%(y)(u(y)− u(x))2 dxdydt

≤ E(%0, u0),(4.5) eq:final


where we have used that f0 = %0(2π)d/2

e−(v−u0)2

2 to conclude the last inequality. Next,we subtract the entropy equality (2.10) from (4.5) to discover

0 ≤ limε→0

∫R2d

fεv2

2+ fε log fε dxdv −

∫Rd%u2

2+ % log % dx ≤ 0, (4.6) ?eq:final2?

where the first inequality follows from (2.5). By convexity of the entropy, we con-clude that

f =%

(2π)d2e−

(u−v)22 ,

which concludes the proof of Theorem 3.1.

4.1. The relative entropy inequality. The fundamental ingredient in the proofof Proposition 4.1 is a relative entropy inequality for the system (1.4) - (1.5) whichwill be derived in this subsection. However, before we embark on the derivation ofthis inequality, we will need some additional identities and simplifications.

First, we recall that E is an entropy due to the existence an entropy flux functionQ such that

djQi(U) =∑k

djAki(U)dkE(U), i, j = 1, . . . , 2. (4.7) def:Q

We then haveE(U)t + divxQ(U) = (−%P + %P )

P

%. (4.8) ?

We note that Q satisfies

divxQ(U) = dE(U) (divxA(U)) . (4.9) eq:Qh

We shall also need the relative flux:

A(V |U) = A(V )−A(U)− dA(U)(V − U),

where the last term is to be understood as

[dA(U)(V − U)]i = dAi(U) · (V − U), i = 1. . . . , d.

The key relative entropy inequality is given by the following proposition:〈prop:realtive entropy〉

Proposition 4.2. Let U =(%%u

)be a strong solution of (4.1) satisfying (2.10)

and let V =(

qQ = qv

)be an arbitrary smooth function. The following inequality

holdsd

dt

∫RdE(V |U) dx

+12

∫Rd

∫RdK(x, y)q(x)q(y) [(v(x)− u(x))− (v(y)− u(y))]2 dxdy

≤∫

Rd

[∂tE(V ) +

12

∫RdK(x, y)q(x)q(y)[v(x)− v(y)]2 dy

]dx

−∫

Rd∇x(dE(U)) : A(V |U) dx

−∫

RddE(U) [Vt + divA(V )− F (V )] dx

+∫

Rd

∫RdK(x, y)q(x)(%(y)− q(y))[u(y)− u(x)][v(x)− u(x)] dxdy.

Remark 4.3. When K = 0, such an inequality was established by Dafermos [10] forgeneral system of hyperbolic conservation laws.


In order to prove Proposition 4.2, we will need the following lemma (see Dafermos[10]).

〈lem:dafermos〉Lemma 4.4. The following integration by parts formula holds∫Rdd2E(U) (divxA(U)) (V − U) dx

=∫

Rd(dA(U)(V − U)) : (∇xdE(U)) dx,

(4.10) magic

where : is the scalar matrix product.

Proof of Lemma 4.4. The proof of this equality can be found at several places in theliterature. For the sake of completeness we recall its derivation here (we follow[2][p. 1812]).

Differentiating (4.7) with respect to Ul, we obtain the identityd∑k=1

dldkE(U)djAki(U) = dldjQi(U)−d∑k=1

dldjAki(U)dkE(U).

Using this identity, we calculate

d2E(U) (divxA(U)) (V − U)

=d∑

lij=1

(d∑k=1

dldkE(U)djAki

)∂Uj∂xi

(Vl − Ul)

=d∑

lij=1

dldjQi(U)∂Uj∂xi

(Vl − Ul)−d∑

lijk=1

dldjAki(U)dkE(U)∂Uj∂xi

(Vl − Ul)

=∑l

div(dlQ(U)(Vl − Ul))−∑kl

divx (dlAk(U)(Vl − Ul)) dkE(U)

+∑l

(∇xVl −∇xUl) ·

[∑k

dlAk(U)dkE(U)− dlQ(U)

].

Now, we observe that (4.7) implies that the last term is zero. Thus, we can conclude

d2E(U) (divxA(U)) (V − U)

= divx (dQ(U)(V − U))− divx (dA(U)(V − U)) dE(U)

Integrating this identity over Rd yields (4.10).

Proof of Proposition 4.2. First of all, we recall that

E(V |U) = E(V )− E(U)− dE(U)(V − U).

We deduce:d

dt

∫RdE(V |U) dx =

∫Rd∂tE(V )− dE(U)Ut − d2E(U)Ut(V − U)

− dE(U)(Vt − Ut) dx

=∫

Rd∂tE(V )− d2E(U)Ut(V − U)− dE(U)Vt dx

:= I1 + I2 + I3.

(4.11) begin

Now, formula (4.10) provides

I2 =∫

Rdd2E(U) [divxA(U)− F (U)] (V − U) dx

=∫

Rd∇xdE(U) : dA(U)(V − U) dx−

∫Rdd2E(U)F (U)(V − U) dx.


By adding and subtracting, and integrating by parts, we find

I3 = −∫

RddE(U) [Vt + divA(V )− F (V )] dx

−∫

Rd(∇xdE(U)) : A(V ) + dE(U)F (V ) dx

Consequently,

I2 + I3 = −∫

Rd∇x(dE(U)) : A(V |U) + dE(U) [Vt + divA(V )− F (V )] dx

−∫

Rd∇x(dE(U)) : A(U) dx

−∫

Rdd2E(U)F (U)(V − U) + dE(U)F (V ) dx := J1 + J2 + J3.

(4.12) i2pi3

Now, using (4.9), we get:

J2 = −∫

Rd∇x(dE(U)) : A(U) dx

=∫

RddE(U) divxA(U) dx =

∫Rd

divxQ(U) dx = 0.(4.13) i2iszero

It only remains to compute the term J3, which requires special attention as itincludes all the contributions of the forcing term F (U). We now insert our specificexpression of E and U . A simple computation yields:

dE(U) =(d%E(U)dPE(U)

)=

(− P 2

2%2 + log %+ 1P%

), d2E(U) =

(∗ − P

%2

− P%2

1%

).

Let us also introduce two functions (q,Q) such that V = [q,Q]T and define v = Qq .

−J3 =∫

Rdd2E(U)F (U)(V − U) + dE(U)F (V ) dx

=∫

Rd− P%2

(%P − %P

)(q − %) +

1%

(%P − %P

)(Q− P )

+P

%

(qQ− qQ

)dx

=∫

Rd

q

%

(%P − %P

)(Qq− P

%

)+P

%

(qQ− qQ

)dx

=∫

Rdq (%u− %u) (v − u) + u

(qQ− qQ

)dx

which yields

−J3 =∫

Rd

∫RdK(x, y)q(x)%(y)[u(y)− u(x)][v(x)− u(x)]

+K(x, y)q(y)q(x)u(x)[v(y)− v(x)] dydx.

=∫

Rd

∫RdK(x, y)q(x)q(y)[u(y)− u(x)][v(x)− u(x)] dxdy

+∫

Rd

∫RdK(x, y)q(x)q(y)[v(y)− v(x)][u(x)− v(x)] dxdy

+∫

Rd

∫RdK(x, y)q(x)q(y)[v(y)− v(x)]v(x) dxdy

+∫

Rd

∫RdK(x, y)q(x)(%(y)− q(y))[u(y)− u(x)][v(x)− u(x)] dxdy.

(4.14) J3


Using the symmetry of K, we see that the first two terms can be written∫Rd

∫RdK(x, y)q(x)q(y)[u(y)− u(x)][v(x)− u(x)] dxdy

+∫

Rd

∫RdK(x, y)q(x)q(y)[v(y)− v(x)][u(x)− v(x)] dxdy (4.15) J31

=12

∫Rd

∫RdK(x, y)q(x)q(y)[u(y)− u(x)][(v(x)− u(x))− (v(y)− u(y))] dxdy

+12

∫Rd

∫RdK(x, y)q(x)q(y)[v(y)− v(x)][(u(x)− v(x))− (u(y)− v(y))] dxdy

=12

∫Rd

∫RdK(x, y)q(x)q(y) [(v(x)− u(x))− (v(y)− u(y))]2 dxdy.

From the symmetry of K, we also easily deduce∫Rd

∫RdK(x, y)q(x)q(y)[v(y)− v(x)]v(x) dxdy

= −12

∫Rd

∫RdK(x, y)q(x)q(y)[v(x)− v(y)]2 dxdy.

(4.16) J32

Hence, by setting (4.16) and (4.15) in (4.14), we discover

J3 =12

∫Rd

∫RdK(x, y)q(x)q(y)[v(x)− v(y)]2 dxdy

− 12

∫Rd

∫RdK(x, y)q(x)q(y) [(v(x)− u(x))− (v(y)− u(y))]2 dxdy

+∫

RdK(x, y)q(x)(%(y)− q(y))[u(y)− u(x)][v(x)− u(x)] dxdy

We conclude the proof by combining the previous identities (4.13), (4.12) and (4.11).

In our proof Proposition 4.1, we will use the following immediate corollary ofProposition 4.2:

?〈cor:ree〉? Corollary 4.5. Let fε be a weak solution of (1.3) satisfying (2.9) and let

Uε = (%ε, %εuε), with %ε =∫

Rdfε dv, %εuε =

∫Rdvfε dv.

Let U = (%, %u) be the strong solution of (4.1) satisfying (2.10). Then the followinginequality holds:

d

dt

∫RdE(Uε|U) dx

+12

∫Rd

∫RdK(x, y)%ε(x)%ε(y) [(uε(x)− u(x))− (uε(y)− u(y))]2 dxdy

≤∫

Rd

[∂tE(Uε) +

12

∫RdK(x, y)%ε(x)%ε(y)[uε(x)− uε(y)]2 dy

]dx

−∫

Rd∇x(dE(U)) : A(Uε|U) dx

−∫

RddE(U) [Uεt + divA(Uε)− F (Uε)] dx (4.17) eq:relative entropy eps

+∫

Rd

∫RdK(x, y)%ε(x)(%(y)− %ε(y))[u(y)− u(x)][uε(x)− u(x)] dxdy.

In order to deduce Proposition 4.1 from this corollary, it remains to show that


(1) The first term in the right hand side in (4.17) is of order ε when integratedwith respect to t (Lemma 4.6).

(2) The second term is controlled by the relative entropy itself (in fact we willshow that the relative flux is controlled by the relative entropy, see Lemma4.7).

(3) The third term is of order O(ε) (Lemma 4.8).(4) The last term can be controlled by the relative entropy (Lemma 4.9).

The rest of this paper is devoted to the proof of these 4 points.

4.2. (1) The first term.

〈lem:negative〉Lemma 4.6. Let fε be the weak solution of (1.3) given by Theorem 2.3 and let


Rdfε dv, %εuε =

∫Rdvfε dv.

Then∫ t

0

∫Rd

[∂tE(Uε) +

12

∫RdK(x, y)%ε(x)%ε(y)[uε(x)− uε(y)]2 dy

]dx ds ≤ εC(t),

for all t ≥ 0.

Proof. First, we write∫ t

0

∫Rd∂tE(Uε) ds

=∫

RdE(Uε)(t)− E(U0) dx

=∫

Rd

[E(Uε)(t)−F(fε)(t)

]+[F(fε)(t)−F(f0)

]+[F(f0)− E(U0)

].

The well-preparedness of the initial data gives∫RdF(f0)− E(U0) dx = 0,

and (2.5) implies ∫RdE(Uε)(t)−F(fε)(t) dx ≤ 0.

Finally, we deduce (using (2.9))∫ t

0

∫Rd∂tE(Uε) ds ≤

∫RdF(fε)(t)−F(f0) dx

≤ −12

∫ t

0

∫Rd

∫RdK(x, y)%ε(x)%ε(y)|uε(y)− uε(x)|2 dydxds

+ C(t)ε

which gives the result.

4.3. (2) Control of the relative flux. We note that |∇xdE(U)| is bounded in L∞

by ‖u‖L∞(0,T∗;W 1,∞), ‖∇ log ρ‖L∞ , so the second term in (4.17) will be controlledif we prove the following standard lemma:

〈lem:flux〉Lemma 4.7. The following inequality holds for all U , V :∫Rd|A(V |U)| dx ≤

∫RdE(V |U) dx


Proof. A straightforward computation gives

dA(U)(V − U)

=

(Q− P

− (q−%)%2 P ⊗ P + 1

%P ⊗ (Q− P ) + 1% (Q− P )⊗ P + (q − %)

),

and using the fact that P = %u and Q = qv, we get:

1qQ⊗Q− 1

%P ⊗ P − 1

%P ⊗ (Q− P )− 1

%(Q− P )⊗ P +

(q − %)%2

P ⊗ P

= qv ⊗ v + %u⊗ u− u⊗ qv − qv ⊗ u+ (q − %)u⊗ u= q(v − u)⊗ (v − u).

Since all the other terms in A(·|·) are linear, we deduce

A(V |U) =(

0 0q(v − u)⊗ (v − u) 0

).

This implies ∫Rd|A(V |U)| dx =

∫Rdq|v − u|2 dx ≤

∫RdE(V |U) dx,

which concludes our proof.

4.4. (3) Kinetic approximation.

〈lem:kineticapproximation〉Lemma 4.8. Let U be a smooth function, let fε be a weak solution of (1.3) satis-fying (2.9) and define


Rdfε dv, %εuε =

∫Rdvfε dv.

There exists a constant C depending on T , ‖u‖L∞(0,T∗;W 1,∞) and ‖∇ log ρ‖L∞ suchthat ∣∣∣∣∫ t

0

∫RddE(U) [Uεt + divA(Uε)− F (Uε)] dx

∣∣∣∣ ≤ √εC(t).

Proof. By setting φ := φ(t, x) in (2.6), we see that

%εt + div(%εuε) = 0 in D′([0, T )× Ω). (4.18) ?

Setting φ := vΨ(t, x), where Ψ is a smooth vector field, we find that

(%εuε)t + divx(%εuε ⊗ uε) +∇x%ε

−∫

RdK(x, y)%ε(t, x)%ε(t, y)(uε(x)− uε(y)) dy

= divx∫

Rd(uε ⊗ uε − v ⊗ v + I) fε dv,

(4.19) ?

in the sense of distributions on [0, T )× Ω. Hence, we have that∣∣∣∣∫ t

0

∫RddE(U) [Uεt + divA(Uε)− F (Uε)] dxdt

∣∣∣∣≤∣∣∣∣∫ t

0

∫Rd|∇xdE(U)|

∣∣∣∣∫Rd

(uε ⊗ uε − v ⊗ v + I) fε dv∣∣∣∣ dxdt∣∣∣∣

≤ C∫ t

0

∫Rd

∣∣∣∣∫Rd

(uε ⊗ uε − v ⊗ v + I) fε dv∣∣∣∣ dxdt,

(4.20) ax:1


where the constant C depends on ‖u‖L∞(0,T∗;W 1,∞) and ‖∇ log ρ‖L∞ . To conclude,we have to prove that the righthand side can be controlled by the dissipation. Asin [22], we calculate∫

Rd(uε ⊗ uε − v ⊗ v + I)fε dv

=∫

Rd(uε ⊗ (uε − v) + (uε − v)⊗ v + I) fε dv

=∫

Rduε√fε ⊗

((uε − v)

√fε − 2∇v

√fε)

+ uε ⊗∇vfε

+(

(uε − v)√fε − 2∇v

√fε)⊗ v√fε +∇vfε ⊗ v + Ifε dv.

Using integration by parts, we see that∫Rduε ⊗∇vfε dv = 0,

∫Rd∇vfε ⊗ v dv =

∫Rd−fI dv.

By applying this and the Holder inequality to (4.4) we find∫ t

0

∫Rd

∣∣∣∣∫Rd

(uε ⊗ uε − v ⊗ v + I)fε dv∣∣∣∣ dx ds (4.21) ax:2

≤∫ t

0

(∫Rd

∫Rdfε|v|2 + fε|uε|2 dvdx

) 12

D1(fε)12 ds ≤

√εC(t),

where the last inequality follows from (2.8) - (2.9) and

%ε|uε|2 =∫fεvuε dv ≤

(∫fεv2 dv

) 12(∫

fε|uε|2 dv) 1

2

=(∫

fεv2 dv

) 12 (%ε|uε|2

) 12 .

We conclude by combining (4.20) and (4.21).

4.5. (4) The last term. Finally, we have:

〈lem:theotherguy〉Lemma 4.9. Assuming that U = (%, %u) and V = (q, qv) are such that u ∈L∞(Rd), q, % ∈ L1(Rd), there exists a constant C such that∫

Rd

∫RdK(x, y)q(x)(%(y)− q(y))[u(y)− u(x)][v(x)− u(x)] dxdy

≤ C‖u‖L∞(‖%‖L1 + ‖q‖L1)∫

RdE(V |U) dx

(4.22) ?

Proof. We have∫Rd


≤ 2‖u‖L∞∫

Rd

∫RdK(x, y)q(x) min

1q(y)

,1

%(y)

1/2

|%(y)− q(y)|

max q(y), %(y)1/2 |v(x)− u(x)| dxdy

≤ 2‖u‖L∞(∫

Rd

∫RdK(x, y)q(x) min

1q(y)

,1

%(y)

(%(y)− q(y))2 dxdy

)1/2

(∫Rd

∫RdK(x, y)q(x) max q(y), %(y) |v(x)− u(x)|2 dxdy

)1/2


And so using (4.3) and the fact that K(x, y) ≤ C, we deduce:∫Rd


≤ C‖u‖L∞‖q‖1/2L1 (‖q‖L1 + ‖%‖L1)1/2(∫

Rdp(q|%)(y) dy

)1/2

×(∫

Rdq(x)[v(x)− u(x)]2 dx

)1/2

≤ C‖u‖L∞‖q‖1/2L1 (‖q‖L1 + ‖%‖L1)1/2∫

RdE(V |U) dx.

〈subsec:prop〉 4.6. Proof of Proposition 4.1. We recall that

E(Uε(0)|U(0)) = 0.

For any t ∈ (0, t), we integrate (4.17) over (0, t) and apply Lemmas 4.7, 4.8 and4.9, and , to obtain the inequality∫

RdE(Uε|U)(t) dx

+12

∫ t

0

∫Rd

∫RdK(x, y)%ε(x)%ε(y) [(uε(x)− u(x))− (uε(y)− u(y))]2 dx dy ds

≤∫ t

0

∫Rd∂tE(Uε) +

12

∫RdK(x, y)%ε(x)%ε(y)[uε(x)− uε(y)]2 dy dx ds

+√εC(t) + C

∫ t

0

∫RdE(Uε|U) dx dt.

Lemma 4.6 now implies:∫RdE(Uε|U)(t) dx

+12

∫ t

0

∫Rd

∫RdK(x, y)%ε(x)%ε(y) [(uε(x)− u(x))− (uε(y)− u(y))]2 dxdydt

≤√ε C(t) + C

∫ t

0

∫RdE(Uε|U) dxdt.

which gives (4.5) and hence completes the proof of Proposition 4.1.


Appendix A. Local well-posedness of the Euler-flocking system〈S5〉 The purpose of this appendix is to prove the existence of a local-in-time unique

smooth solution of the Euler-flocking equations. In particular, the objective is toprove Theorem 2.4 which we relied upon to prove our main result (Theorem 3.1).

First, we observe that the system (1.4)-(1.5) is a 4 × 4 system of conservationlaws which can be written in the following equivalent form when the solution issmooth:

∂t%+ u∇x%+ % divx u = 0%(∂tu+ u∇xu) +∇x% = F (%, u,∇xΦ).

(A.1) Euler-smooth

Here,

F (%, u,∇Φ(x)) =∫

RdK(x, y)%(x)%(y)[u(y)− u(x)] dy − %∇xΦ. (A.2) ?potential?

For s > 0, Hs(Rd) denotes the usual Sobolev space, equipped with the norm

‖g‖2s =∫

Rd(1 + |ξ|2s)|g(ξ)|2 dξ.

For g ∈ L∞([0, T ];Hs), we also define

‖|g‖|s,T = sup0≤t≤T

‖g(·, t)‖s.

Now, consider the Cauchy problem of (A.1) with smooth initial data:

(%, u)|t=0 = (%0, u0)(x). (A.3) ID

The values of the vector

w =(%u

)lie in the state space G = (%, u)> : % > 0 which is an open set in R4. Indeed,by invoking the method of characteristics for the continuity equation the followinglemma holds:

Lemma A.1. If (%, u) ∈ C1(R3 × [0, T ]) is a uniformly bounded solution of (1.4)with %(x, 0) > 0, then %(x, t) > 0 on Rd × [0, t].

System (A.1) can thus be written in the form

∂t

(%u

)+ u∇x

(%u

)+(%divx u∇x%%

)=(

01%F (%, u,∇xΦ(x)

)or equivalently

∂tw +∇x · f(w) =(

01%F (%, u,∇xΦ(x)

)(A.4) h-s

where

f(w) =(

%u>

u⊗ u+ log %I3

)where I3 denotes the 3× 3 identity matrix.

It turns out that (A.4) has a structure of symmetric hyperbolic systems. To seethis, we define (for w ∈ G):

A0(w) =(%−1 00 %I3

).

The matrix A0(w) (called the symmetrizing matrix of system (A.4)) is positivedefinite and is smooth with respect to w. Furthermore, it satisfies:

c−10 I4 ≤ A0(w) ≤ c0I4,


with a constant c0 uniform for w ∈ G1 ⊂ G1 ⊂ G.Multiplying (A.4) by A0 we obtain the symmetric hyperbolic system

A0(w)∂tw +A(w)∇xw = G(w,∇xΦ) (A.5) in-h-s

with smooth initial dataw0 = w(x, 0) (A.6) ID-in-h-s

where the columns of the matrix A(w) = (A1(w), A2(w), A3(w)), defined by

Ai(w) = A0(w)Dfi(w)

are symmetric 4× 4 matrices. The 4× 1 vector G in (A.5) is given by

G(w,∇xΦ) = A0(w)(

01%F (%, u,∇xΦ)

)=(

0F (%, u,∇xΦ)

)=

(0∫

Rd K(x, y)%(x)%(y)[u(y)− u(x)] dy − %∇xΦ

). (A.7) ?source?

We are now ready to prove the local existence of smooth solutions.〈local-existence〉Theorem A.2. Assume w0 = (%0, u0) ∈ Hs∩L∞(Rd) with s > d

2 +1 and %0(x) > 0and assume that ∇xΦ ∈ Hs. Then there is a finite time T ∈ (0,∞), depending onthe Hs and L∞ norms of the initial data, such that the Cauchy problem (1.4)-(1.5)and (A.3) has a unique bounded smooth solution w = (%, u) ∈ C1(Rd× [0, T ]), with% > 0 for all (x, t) ∈ Rd × [0, T ], and (%, u) ∈ C([0, t];Hs) ∩ C1([0, T ];Hs−1).

Theorem A.2 is a consequence of the following theorem on the local existence ofsmooth solutions, with the specific state space G = (%, u)> : % > 0 ⊂ R4 for theinhomogeneous system (A.5).

〈local-smooth-in-h-s〉Theorem A.3. Assume that w0 : Rd → G is in Hs∩L∞ with s > d2 +1. Then, for

the Cauchy problem (A.5)-(A.6), there exists a finite time T = T (‖w0‖s, ‖w0‖L∞) ∈(0,∞) such that there is a classical solution w ∈ C1(Rd × [0, T ]) with w(x, t) ∈ Gfor (x, t) ∈ Rd × [0, T ] and w ∈ C([0, T ];Hs) ∩ C1([0, T ];Hs−1).

The proof of this theorem proceeds via a classical iteration scheme method. Anoutline of the proof of Theorem A.3 (and therefore Theorem A.2) is given below.

Proof of Theorem A.3. Consider the standard mollifier

η(x) ∈ C∞0 (Rd), supp η(x) ⊆ x; |x| ≤ 1, η(x) ≥ 0,∫

Rdη(x)dx = 1,

and setηε = ε−dη(x/ε).

Define the initial data wk0 ∈ C∞(Rd) by

wk0 (x) = ηεk ? w0(x) =∫

Rdηεk(x− y)w0(y)dy,

where εk = 2−kε0 with ε0 > 0 constant. We construct the solution of (A.5)-(A.6)using the following iteration scheme. Set w0(x, t) = w0

0(x) and define wk+1(x, t),for k = 0, 2, . . . , inductively as the solutions of linear equations:

A0(wk)∂twk+1 +A(wk)∇xwk+1 = G(wk,∇xΦ),wk+1|t=0 = wk+1

0 (x).(A.8) it-sc

By well-known properties of mollifiers it is clear that:

‖wk0 − w0‖s → 0, as k →∞, and ‖wk0 − w0‖0 ≤ C0εk‖w0‖1,for some constant C0. Moreover, wk+1 ∈ C∞(Rd × [0, Tk]) is well-defined on thetime interval [0, Tk], where Tk > 0 denotes the largest time for which the estimate


‖|wk − w00‖|s,Tk ≤ C1 holds. We can then assert the existence of T∗ > 0 such that

Tk ≥ T∗ (T0 =∞) for k = 0, 1, 2, . . . , from the following estimates:

‖|wk+1 − w00‖|s,T∗ ≤ C1, ‖|wk+1

t ‖|s−1,T∗ ≤ C2, (A.9) est

for all k = 0, 1, 2, . . . , for some constant C2 > 0. From (A.8), we get

A0(wk)∂t(wk+1 − wk) +A(wk)∇(wk+1 − wk) = Ek +Gk, (A.10) lin-bl

whereEk = −(A0(wk)−A0(wk−1)∂twk − (A(wk)−A(wk−1))∇wk,Gk = G(wk+1,∇xΦ)−G(wk,∇xΦ).

(A.11) ?

From the standard energy estimate for the linearized problem (A.10) we get

‖|wk+1 − wk‖|0,T ≤ CeCT (‖wk+10 − wk0‖0 + T‖|Ek‖|0,T + T‖|Gk‖|0,T ).

Taking into consideration the property of mollification and relation (A.9), we get

‖wk+10 − wk0‖0 ≤ C2−k, ‖|Ek‖|0,T ≤ C‖|wk − wk−1‖|0,T

and

‖|Gk‖|0,T = sup0≤t≤T

‖Gk(·, t)‖0 = sup0≤t≤T

‖G(wk+1,∇xΦ)−G(wk,∇xΦ)‖0

≤ C‖wk+1 − wk‖0,Twith C depending on the constant appearing in (A.9) and on ||K||L∞ . We deduce:

‖|wk+1 − wk‖|0,T ≤ CeCT (2−k + T‖|wk − wk−1‖|0,T + T‖|wk+1 − wk‖|0,T ).

Choosing T small enough such that CTeCT

1−CTeCT < 1 one obtains∞∑k=1

‖|wk+1 − wk‖|0,T <∞,

which implies that there exists w ∈ C([0, T ];L2(Rd)) such that

limk→∞

‖|wk − w‖|0,T = 0. (A.12) conv

From (A.9), we have ‖|wk‖|s,t + ‖|wkt ‖|s−1,T ≤ C, and wk(x, t) belongs to a boundedset of G for (x, t) ∈ Rd × [0, T ]. Then by interpolation we have that for any r with0 ≤ r < s,

‖|wk − wl‖|r,T ≤ Cs‖|wk − wl‖|1−r/s0,T ‖|wk − wl‖|r/ss,T ≤ ‖|wk − wl‖|1−r/s0,T . (A.13) last

From (A.12) and (A.13),limk→∞

‖|wk − w‖|r.,T = 0

for any 0 < r < s. Therefore, choosing r > 32 + 1, Sobolev’s lemma implies

wk → w in C([0, t];C1(Rd)). (A.14) cnv

From (A.10) and (A.14) one can conclude that wk → w in C([0, T ];C(Rd), w ∈C1(Rd × [0, T ]), and w(x, t) is a smooth solution of (A.5)-(A.6). To prove u ∈C([0, T ];Hs) ∩ C1([0, T ];Hs−1), it is sufficient to prove u ∈ C([0, T ];Hs) since itfollows from the equations in (A.5) that u ∈ C1([0, T ];Hs−1).

Remark A.4. The proof of Theorem A.2 follows the line of argument presentedby Majda [21], which relies solely on the elementary linear existence theory forsymmetric hyperbolic systems with smooth coefficients (see also Courant-Hilbert[5]).

Finally, we have the following lemma:


〈lem:〉Lemma A.5. Let (%, u) be a sufficiently smooth solutions to (1.4)-(1.5). Then thefollowing entropy equality holds:

∂tE (ρ, u) +12

∫Rd

∫RdK(x, y)%(x)%(y) |u(x)− u(y)|2 dydx = 0. (A.15) en-e

Proof. Equality (A.15) follows from a straightforward computation, multiplying thecontinuity equation (1.11) by (% log %)′ and the momentum equation (1.12) by thevelocity field u, and adding the resulting relations.

A.1. Proof of Theorem 2.4. Theorem 2.4 now follows from Theorem A.2 andLemma A.5.

Appendix B. Entropy inequality

We recall that the entropy is given by

F(f) =∫

R2df log f + f

v2

2dv dx

and the associated dissipations by

D1(f) =∫

R2d

1f|∇vf − f(u− v)|2 dv dx

and

D2(f) =12

∫Rd

∫Rd

∫Rd

∫RdK0(x, y)f(x, v)f(y, w) |v − w|2 dwdydvdx

The first part of Proposition 2.1 follows from the following lemma:

?〈lem:entropyeq〉? Lemma B.1. Let f be a sufficiently integrable solution of (1.3), then

∂tF(f) +1εD1(f) +D2(f)

= d

∫Rd

∫Rd

∫Rd

∫RdK0(x, y)f(x, v)f(y, w) dwdydvdx. (B.1) eq:entropy1’

Proof. Using the equation (1.3), we calculate

∂tF(f) =∫

Rd

∫Rdft (log f + v) dvdx

=∫

Rd

∫RdL[f ]∇vf −

1f

1ε

(∇f − f(u− v))∇vf dvdx

+∫

Rd

∫RdfL[f ]v − 1

ε(∇f − f(u− v)v) dvdx

=∫

Rd

∫Rd−f divv L[f ] + vfL[f ] dvdx

−∫

Rd

∫Rd

1f

1ε

(∇f − f(u− v)) (∇vf + vf) dvdx := I + II.

(B.2) eq:st1


By definition of L[f ], we deduce

I :=∫

Rd

∫Rd−f divv L[f ] + vfL[f ] dvdx

=∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(x, v)f(y, w) divv v dwdydvdx

+∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(x, v)f(y, w)(w − v)v dwdydvdx

= d

∫Rd

∫Rd

∫Rd

∫RdK0(x, y)f(x, v)f(y, w) dwdydvdx−D2(f),

(B.3) eq:I

where we have used the symmetry of K0(x, y)f(x)f(y) to conclude the last equality.By adding and subtracting u, we rewrite II as follows:

II : = −1ε

∫Rd

∫Rd

1f

(∇vf − f(u− v)) (∇vf + vf) dvdx.

= −1εD1(f) +

1ε

∫Rd

∫Rd−u∇vf + fu(u− v) dvdx (B.4) eq:II

= −1εD1(f) +

1ε

∫Rd%u2 − %u2 dx = −1

εD1(f).

We conclude by setting (B.3) and (B.4) in (B.2).

To prove the second inequality (2.4) in Proposition 2.1, we must prove that theright-hand side in (B.1) can be controlled by the dissipation and the entropy. Wewill need the following lemma:

〈lem:confinement〉Lemma B.2. For all T , there exists a constant C(T ) such that if f is a smoothsolution of (1.3), then

F(f) +∫

R2dfx2

2dvdx ≤

(F(f0) +

∫R2d

f0x2

2dvdx

)eC(T ), (B.5) ent:x

for all t ≤ T . Furthermore, we then have(∫Rd

∫Rdf log+ f + f

(v2

4+x2

4

)dvdx

)(t)

+∫ t

0

1εD1(f) +D2(f) ds ≤ C(t)

(B.6) eq:entroconf

Proof. We write

F(f) =∫

R2df log+ f − f log− f + f

v2

2dv dx

where log+(s) = max0, log(s) and log−(s) = max0,− log(s).Using the previous lemma, we have that

d

dt

(F(f) +

∫R2d

fx2

2dvdx

)≤∫

R2dfvx dvdx+ d

∫R4d

K0(x, y)f(y, w)f(x, v) dwdydvdx

≤∫

R2dfv2

2+ f

x2

2+ 2f log− f − 2f log− f dvdx

+ d

∫R4d

K0(x, y)f(y, w)f(x, v) dwdydvdx.

(B.7) kind-of-weird


To proceed, we shall need the classical inequality (cf. [13]):

2∫

R2df log− f dvdx

≤∫f

(x2

2+v2

2

)dvdx+

1e

∫R2d

e−v24 −

x24 dvdx

(B.8) neg-log

By applying this in (B.7), we obtain

d

dt

(F(f) +

∫R2d

fx2

2dvdx

)≤ 2

(F(f) +

∫R2d

fx2

2dvdx

)+

1e

∫R2d

e−v24σ−

x24σ dvdx

+ d

∫R4d

K0(x, y)f(y, w)f(x, v) dwdydvdx.

Since K0 is bounded, an application of the Gronwall inequality gives (B.5).Finally, (B.6) is a consequence of (B.5), the previous lemma, and (B.8).

Inequality (2.4) now follows from the following lemma:

〈lem:weirdtermiscontrolled〉Lemma B.3. For all T , there is a constant C > 0, depending only on T , ‖K0‖∞,and the total mass M , such that

12

∫Rd

∫RdK0(x, y)%(x)%(y) |u(x)− u(y)|2 dydx ≤ D2(f) +

12εD1(f)

− d∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(x, v)f(y, w) dwdydvdx+ εC(T )

or all t ≤ T .

Proof of Lemma B.3. By symmetry of K0(x, y), we have

12

∫Rd

∫RdK0(x, y)%(x)%(y) |u(x)− u(y)|2 dydx

=∫

Rd

∫RdK0(x, y)%(x)%(y) (u(x)− u(y))u(x) dydx

=∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(x, v)f(y, w) (v − w)u(x) dwdydvdx

=∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(y, w) (v − w)

× (f(x, v)(u(x)− v)−∇vf(x, v)) dwdydvdx

+∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(y, w)f(x, v) (v − w) v dwdydvdx

+∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(y, w) (v − w)∇vf(x, v) dwdydvdx

= I + II + III.

(B.9) eq:st

Let us first consider the last term. Integration by parts provides the identity

III =∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(y, w) (v − w)∇vf(x, v) dwdydvdx

= −d∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(y, w)f(x, v) dwdydvdx.

(B.10) eq:III2


By symmetry of the kernel K0(x, y), we have that

II =∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(y, w)f(x, v) (v − w) v dwdydvdx

=∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)f(y, w)f(x, v)

|v − w|2

2dwdydvdx.

(B.11) ?eq:II2?

It remains to bound I. For simplicity, let us introduce the notation

V (x, v) =1√

f(x, v)(f(x, v)(u(x)− v)−∇vf(x, v)) .

Using this notation, some straight forward manipulations, and the Holder inequal-ity, we obtain using Lemma B.2,

I =∫

Rd

∫Rd

∫Rd

∫RdK0(x, y)

√f(x, v)f(y, w) (v − w)V (x, v) dwdydvdx

=∫

Rd

∫Rd

∫RdK0(x, y)

√f(x, v)%(y)(v − u(y))V (x, v) dydvdx

=∫

Rd

(∫RdK0(x, y)%(y) dy

)∫Rdv√f(x, v)V (x, v) dvdx

−∫

Rd

(∫RdK0(x, y)%(y)u(y) dy

)∫Rd

√f(x, v)V (x, v) dvdx

≤ ‖K‖L∞M(∫

Rd

∫Rd|v|2f(x, v) dvdx

) 12(∫

Rd

∫Rd|V (x, v)|2 dvdx

) 12

+ ‖K‖L∞M12

(∫Rdfv , dv dx

)(∫Rd

∫Rd|V (x, v)|2 dvdx

) 12

≤ 12εD1(f) + ε‖K‖2L∞M2

∫Rd

∫Rd|v|2f(x, v) dvdx.

(B.12) eq:I2

We conclude the result by setting (B.10) - (B.12) in (B.9) and using (B.6).

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(Karper)Center for Scientific Computation and Mathematical Modeling, University of Mary-

land, College Park, MD 20742

E-mail address: [email protected]

URL: folk.uio.no/~trygvekk

(Mellet)

Department of Mathematics, University of Maryland, College Park, MD 20742E-mail address: [email protected]

URL: math.umd.edu/~mellet

(Trivisa)Department of Mathematics, University of Maryland, College Park, MD 20742

E-mail address: [email protected]

URL: math.umd.edu/~trivisa

http://folk.uio.no/~trygvekk

http://www.math.umd.edu/~mellet

http://www.math.umd.edu/~trivisa

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