long-time behaviour of nonlinear fokker-planck equations · 2018-07-03 · formulae, we consider...

Angewandte Mathematik

Long-Time Behaviour of Nonlinear

Fokker-Planck Equations

Angewandte Mathematik

Long-Time Behaviour of Nonlinear

Fokker-Planck Equations

Diplomarbeit

vorgelegt von

Jan-Frederik Pietschmann

Westfalische Wilhelms-Universitat Munster

Institut fur Numerische und Angewandte Mathematik

— November 2008 —

Contents

1 Introduction 5

1.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 The Optimal Transportation Problem 9

2.1 Monge’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Kantorovich’s Approach . . . . . . . . . . . . . . . . . . . . . 11

2.3 Duality formulation . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Quadratic Costs . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Kantorovich-Wasserstein distances . . . . . . . . . . . . . . . 14

2.5.1 Benamou-Brenier Reformulation . . . . . . . . . . . . 15

2.6 Displacement Convexity . . . . . . . . . . . . . . . . . . . . . 18

3 Gradient Flow Formulations of PDEs 21

3.1 Introduction to Gradient Flows . . . . . . . . . . . . . . . . . 21

3.2 Otto’s Approach for the porous medium equation . . . . . . . 23

3.2.1 Gradient Flow Formuation . . . . . . . . . . . . . . . 24

3.2.2 Formal Derivation of Convergence Rates to StationarySolutions . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Gradient flow for Systems . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.2 Gradient flow interpretation . . . . . . . . . . . . . . . 28

3.3.3 Distance and Geodesics on (M, g) . . . . . . . . . . . 31

4 Entropy Dissipation Methods 35

4.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3

4 CONTENTS

5 Application to Systems of Diffusive PDEs 39

5.1 Entropy Dissipation Methods . . . . . . . . . . . . . . . . . . 395.1.1 The Scalar Case . . . . . . . . . . . . . . . . . . . . . 40

5.2 Gradient Flow and Displacement Convexity . . . . . . . . . . 445.2.1 Linear Scalar Case . . . . . . . . . . . . . . . . . . . . 445.2.2 Nonlinear Scalar Case with Potential . . . . . . . . . . 465.2.3 System Case without Potential . . . . . . . . . . . . . 505.2.4 Chen Model for Energy Transport . . . . . . . . . . . 51

5.3 Purely One-Dimensional Calculations . . . . . . . . . . . . . 535.3.1 Entropy Dissipation in One Dimension . . . . . . . . . 535.3.2 Displacement Convexity in One Dimension . . . . . . 55

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Summary 57

A Some facts from Measure Theory 59

A.1 Convex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 60

B Some facts from Differential Geometry 61

B.1 The Tangent Space . . . . . . . . . . . . . . . . . . . . . . . . 63B.2 Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . 63

Bibliography 64

Chapter 1

Introduction

This thesis deals with the convergence of solution of a system of non-linearFokker-Planck equations

ρt = div(D(ρ)(∇u′(ρ) +∇V (x))), x ∈ RN , t ≥ 0 (1.1)

to a equilibrium or stationary state. These are defined as solutions whichdo not depend on time anymore. The Fokker-Planck equation plays a keyrole in many areas of physics. The linear Fokker-Planck equation

ρt = div(∇ρ+ ρ∇V (x)), x ∈ RN , t ≥ 0

comes from the underlying stochastic differential equation

dXt = dWt −∇V (Xt)dt,

where Wt is a Brownian motion (also called standard Wiener process). Thisrepresentation makes clear that Fokker-Planck equations describe the dy-namics of a set of particles that are influenced by both diffusion and drift.For a comprehensive review see [Risken89].For the linear equation a stationary state, which is defined as a solutionwhich does not change in time, is given by

ρ = e−V .

Given this state, it can be asked if all solutions of the equation converge tothis state as t→∞ and if they do, at which rate.

5

6 Chapter 1. Introduction

To explore this behaviour for equation (1.1), two techniques are used.The first one is the so-called entropy dissipation method. Here, an entropyfunctional is definied and then used to controll the distance between a so-lution at time t and the steady state. By studying the derivatives of thisfunctional and using log-sobolev inequalities as well as Gronwall’s inequality,it is possible to derive rates for the convergence in terms of relative Entropy.These can be translated to rates in L1 using Czisar-Kullback type inequali-ties.The second aprroach is to see the Fokker-Planck equation as a Gradientflow, that means as a dynamical system that evolves according to

ρt = −gradE(ρ).

The idea is to define a manifold such that the Fokker-Planck equation can beunderstood as such a system on this manifold. If it is furthermore possibleto show that the functional E is convex, a contraction principle follows.The proper sense of convexity here is the so-called displacement convexitywhich reveals to connection to the Monge-Kantorovich problem of optimaltransport.

1.1 Acknowledgements

First of all I would like to thank Prof. Martin Burger and Prof. PeterMarkowich for giving me the possibility to write a thesis on this interestingtopic and for many helpful discussions. Furthermore, I would like to thankthe Wolfgang Pauli Institut (WPI) in Vienna for providing financial supportfor a four week stay at Cambridge this February. Also, I would like to thankthe Institute for Pure and Applied Mathematics (IPAM), which allowed meto participate in the Optimal Transport long-term program and to stay inLos Angeles for three and a half months. During this time, I had a numberof useful discussions and I especially want to thank Jose Antonio Carrillode la Plata and Yann Brenier for their help. Also, I want to thank AlessioFigalli and Marcus Wunsch for reading parts of this thesis and ChristianDoring for delivering it. Finally, I’d like to thank my parents and of courseMaja for their support over the years.

1.2. Notation 7

1.2 Notation

Throughout this thesis, we will use the following notations. Let v ∈ RN bea real-valued vector and A ∈ RN×N be a real-valued matrix. We will usethe following notations

• By Ai, we denote the i-th row, by Ak the k-th column of A.

• The (vector-valued) divergence operator will act row-wise on matrices,i.e.

divA(x) = (divA1, . . . ,divAN ).

• For the scalar product of two matrices we write

A : B = tr(ABT ) =N∑i=1

Ai ·Bi,

where · denotes the scalar product in RN .

• The gradient of a matrix is defined as the matrix which has as entriesthe gradient of each component of the original matrix, i.e.

∇A =

∇A11 · · · ∇A1N

.... . .

...∇AN1 · · · ∇ANN

• The Jacobian matrix of a vector map x 7→ v(x), x ∈ RN has gradients

on its rows, i.e.

∇v(x) = [∇v1(x), . . . ,∇vN (x)] .

Furthermore, we recall the following theorem:

Theorem 1.2.1 (A Green’s formula for matrices) Let RN 3 x 7→ V (x) ∈RN and RN 3 x 7→ A(x) ∈ RN×N be smooth maps. Let Ω be a bounded sub-domain of RN with smooth boundary ∂Ω having x 7→ ν(x) as normal unitvector. Then,∫

ωV (x)TdivA(x)dx =

∫∂ΩV (x)TA(x)ν(x)dσ −

∫Ω∇V (x) : A(x)dx.

8 Chapter 1. Introduction

Chapter 2

The Optimal Transportation

Problem

The problem of optimal transportation was originally proposed by Mongein 1781 in ”memoire sur la theorie des deblais et des remblais”. Today thereexists a vast literature about this topic. We particularly mention the sur-veys by Evans [Evans01] and Ambrosio [Ambrosio00] as well as the book byVillani [Villani03].

2.1 Monge’s Problem

The original problem of Monge was how to transfer a pile of sand into a holewith the lowest amount of work possible. To express this in mathematicalformulae, we consider two separable metric space X and Y and denote byP(X), P(Y ) the set of all probability measures on X, Y . We then take twoprobability measures µ ∈ P(X) and ν ∈ P(Y ), both having the same finitemass,

µ(X) = ν(Y ) <∞. (2.1)

Furthermore, we make the following definition [Ambrosio05].

Definition (push-forward) LetX, Y be separable metric spaces, µ ∈ P(X),and r : X → Y is a µ-measurable map denoted by r#µ ∈ P(Y ). The push-forward of µ through r is defined by

r#µ(B) := µ(r−1(B)) ∀B ∈ B(Y ), (2.2)

9

10 Chapter 2. The Optimal Transportation Problem

where B(Y ) is the family of all Borel subsets of Y . More generally we have∫Xf(r(x))dµ(x) =

∫Yf(y)dr#µ(y) (2.3)

for every bounded (or r#µ-integrable) Borel function f : X → R.

Given a Borel cost function c : X × Y → [0,+∞] that describes the cost ofmoving mass from location x to location y as c(x, y), the Monge problem isgiven by

inft

∫Xc(x, t(x))dµ(x) : t#µ = ν

. (2.4)

In the original formulation, Monge was interested in the case that the work isproportional to the distance, i.e. c(x, y) = |x−y|. Other examples would bethe distance squared (c(x, y) = |x−y|2) which leads to interesting theoreticalresults or the relativistic case which is

c(x− y) =

1−

√1− |x− y|2 if |x− y| ≤< 1

+∞ else.

Existence and Uniqueness of Solutions

In the following, we state two examples from [Ambrosio00] that show thatneither existence nor uniqueness for the general problem can be expected.

Example (Non-existence) Let µ = δ0 be the Dirac delta-measure locatedat zero and ν = (δ−1/2 + δ+1/2)/2. Then there exists no Solution to (2.4),because there is no map t such that t#µ = ν. In other words: In Monge’sProblem does not allow the splitting of mass.

To illustrate non-uniqueness, the cost function c(x, y) = |x−y| is considered.In this case, it can be proven that the infimum Monge’s problem is alwaysgreater than ∑∫

Xud(ν − µ) : u ∈ Lip1(X)

. (2.5)

We use this for the following example.

Example (Non-uniqueness) Let n ≥ 1 be an integer and µ = χ[0,n]L1 andν = χ[1,n+1]L1. Then the map ψ(t) = t + 1 is optimal. Indeed, the costrelative to ψ is n and choosing the 1-Lipschitz function u(t) = t in (2.5), we

2.2. Kantorovich’s Approach 11

obtain that the supremum is at least n, whence the optimality of ψ follows.But since the optimal cost is n, if n > 1 another optimal map ψ is given by

ψ(t) =

t+ n on [0, 1]

t on [1, n]. (2.6)

However, under additional assumptions on the cost function c, existence anduniqueness of an optimal map can be ensured. For example, if µ, ν ∈ P(RN )and c is of the form c(x, y) = h(x−y) with h : RN → [0,+∞) strictly convex,then there exist a unique solution to (2.4). (See [McCann95].)

2.2 Kantorovich’s Approach

Kantorovich’s approach is a relaxed formulation of Monge’s Problem. In-stead of looking for transport maps, we are interested in transport planswhich are probability measures on the product space X × Y . We are onlyusing transport plans π ∈ P(X × Y ) whose marginals are the measures µand ν, i.e.

π[A× Y ] = µ[A], π[X ×B] = ν[B], (2.7)

for all measurable subsets A ⊂ X and B ⊂ Y . We denote the set of all suchtransport plans by Π(µ, ν). Then the Kantorovich problem is given by

minπI[π], (2.8)

withI[π] := min

π

∫X×X

c(x, y)dπ(x, y) : π ∈ Π(µ, ν)

(2.9)

2.3 Duality formulation

It is known that a linear minimization problem with convex constraintshas a dual formulation. For the optimal transport problem in Kantorovichformulation this was introduced by Kantorovich himself in 1942.

Theorem 2.3.1 Under the assumption that c is lower semi-continous, theminimum of the Kantorovich problem is equal to

supJ(ϕ,ψ), (2.10)


withJ(ϕ,ψ) := sup

∫Xϕ(x)dµ(x) +

∫Yψ(y)dν(y)

, (2.11)

where the supremum is taken over all pairs (ϕ,ψ) ∈ C0b (X) × C0

b (Y ) suchthat ϕ(x) + ψ(y) ≤ c(x, y). The set of all (ϕ,ψ) fullfilling this condition isdenoted by Φc.

Proof (formal) To justifiy this theorem, we state a formal proof from Villani[Villani03]. We write the constrained infimum problem as a inf sup problemand then use some minmax principle to replace ”inf sup” by ”sup inf”. Firstwe introduce the indicator function of Π as the function

1Π(π) =

0 if π ∈ Π(µ, ν)

+∞ else(2.12)

Since all the constraints defining Π are linear, this can be written as thesolution of a supremum problem involving only linear functionals. This is

1Π(π) = sup(φ,ψ)

[∫φdµ+

∫ψdν −

∫[φ(x) + ψ(y)]dπ(x, y)

],

for all (φ, ψ) ∈ Cb(X) × Cb(Y ). Using the following formulation of theKantorovich problem,

infπ∈Π(µ,ν)

I[π] = infπ∈P(µ,ν)

(I[π] + 1Π(π)) , (2.13)

we get

infπ∈Π(µ,ν)

I[π] = infπ∈P(µ,ν)

sup(φ,ψ)

∫X×Y

c(x, y)dπ(x, y)

+[∫

φdµ+∫ψdν −

∫[φ(x) + ψ(y)]dπ(x, y)

],

because the first integral does not depend on ϕ and ψ. If now we assume aminmax principle that allows us to interchange sup and inf, then

infπ∈Π(µ,ν)

I[π] = supφ,ψ

∫φdµ+

∫ψdν

− supπ∈P(µ,ν)

∫X×Y

[φ(x) + ψ(y)− c(x, y)] dπ(x, y)

.

2.4. Quadratic Costs 13

We now want to calculate the second supremum over all π ∈ P(µ, ν) .Therefore we define ξ(x, y) := φ(x) + ψ(x)− c(x, y) and consider two cases:If ξ has a positive value at some point (x0, y0), then by choosing π = λδ(x0,y0)

and letting λ→ +∞, we see that the supremum in infinity. If, on the otherhand, ξ ≤ 0 for all (x, y) (dµ ⊗ dν-everywhere), when the supremum isobtained for π = 0. Therefore,

supπ∈P(µ,ν)

∫X×Y

[φ(x) + ψ(y)− c(x, y)] dπ(x, y)dπ(x, y) =

0 if (φ, ψ) ∈ Φc

+∞ else(2.14)

holds. By substituting this formula into (2.14), we finally obtain

infπ∈Π(µ,ν)

I[π] = supφ,ψ∈Φc

∫φdµ+

∫ψdν

= sup

(ϕ,ψ)∈Φc

J(ϕ,ψ). (2.15)

2.4 Quadratic Costs

For the special case of a cost function of the form c(x, y) = |x − y|2 thereexist additional results. Especially popular is the following one, which in[Villani03] is referred as Brenier’s theorem.

Remark Here µ does not give mass to small sets means that µ does notgive mass to sets of at most Hausdorff dimension n− 1.

Theorem 2.4.1 Let µ, ν be probability measures on RN with finite secondmoments. We consider the Monge-Kantorovich problem with the quadraticcost function. Then, if µ does not give mass to small sets, there is a uniqueoptimal π which is

dπ(x, y) = dµ(x)δ∇ϕ(x)(y), (2.16)

or equivalently,π = (Id×∇ϕ)#µ, (2.17)

where ∇ϕ is the unique (i.e. uniquely determined dµ-almost everywhere)gradient of a convex function which pushes µ forward to ν. Moreover,

Supp(ν) = ∇ϕ(Supp(µ)). (2.18)


Under certain assumptions, it is also possible to derive a differential for-mulation of Monge’s Problem. We consider two measures µ, ν ∈ P(RN ),absolutely continous with respect to the Lebesgue measure. Then, accordingto the Radon-Nikodym theorem, we can write

dµ = f(x)dx, dν = g(x)dx, (2.19)

for some densities f , g ∈ L1(RN ). Using the definition of the push-forward,the mass transportation problem is∫

ξ(y)g(y)dy =∫ξ(∇ϕ(x))f(x)dx, (2.20)

for all test functions ξ ∈ Cb(RN ). If we further assume that ∇ϕ is smoothand one-to-one, we can apply the change of variables formula with y =∇ϕ(x) ∫

ξ(y)g(y)dy =∫ξ(∇ϕ(x))g(∇ϕ(x))|det(D2φ(x))|dx. (2.21)

Therefore, as ξ is an arbitrary test function, we get (neglecting the absolutevalue because D2ϕ > 0 due to the convexity of ϕ)

f(x) = g(∇ϕ(x)) det(D2ϕ(x)) (2.22)

or, if g is positive,

det(D2ϕ(x)) =f(x)

g(∇ϕ(x)). (2.23)

This equation is usually called Monge-Ampere equation.

2.5 Kantorovich-Wasserstein distances

Definition (Kantorovich-Wasserstein distances) Given a metric space (X, d)the p-Kantorovich-Wasserstein distance between two measures in the spaceP(X) is definied as

Wp(µ, ν) =(

infπ∈Π(µ,ν)

∫Xd(x, y)pdπ(x, y)

)1/p

(2.24)

2.5. Kantorovich-Wasserstein distances 15

2.5.1 Benamou-Brenier Reformulation

In their paper [Benamou00], Benamou and Brenier provide an alternativeformulation for the L2 Kantorovich-Wasserstein distance, in order to pro-vide a numerical solution. Given a fixed time intervall [0, T ], they transferthe problem to a continuum mechanical framework. Considering every-thing smooth enough, time-dependent density and velocity fields ρ(x, t) ≥ 1,v(x, t) ∈ RN , t ∈ [0, T ] and x ∈ RN subject to the continuity equation

ρt + div(ρv) = 0, (2.25)

and the initial and final conditions

ρ(0, ·) = ρ0 ρ(T, ·) = ρT . (2.26)

In this setting, they prove the following proposition.

Proposition 2.5.1 The square of the L2 Kantorovich distance is equal tothe infimum of

T

∫RN

∫ T

0ρ(t, x)|v(t, x)|2dxdt, (2.27)

among all (ρ, v) satisfying (2.25) and (2.26).

Proof The proof of this proposition is obtained using Lagrangian coordi-nates. We assume that ρ0 and ρT are compactly supported in RN andbounded. Furthermore, we consider sufficiently smooth field ρ, v satisfying(2.25) and (2.26). This again makes this proof only formal. It can be maderigorous using for example approximation arguments. We will state partof the proof from [Villani03, Section 8.1]. Another version can be found in[Ambrosio05, Section 8.1].To introduce Lagrangian coordinates, we define X(t, x) by

X(0, x) = x,d

dtX(t, x) = v(t,X(t, x)), (2.28)

and assume that (X(t, ·))0≤t≤T is a locally Lipschitz family of diffeomor-phisms of RN .

Step I:

First of all, we show that the push forward (X(t, ·))#ρ0 is a solution of the


continuity equation in the distributional sense. Again, we neglect all issueslike existence of derivatives, etc.Let µ be a probability measure on RN , and ρ(t, ·) = X(t, ·)#µ. We willshow that this is a solution of (2.25) in C([0, T ];P (RN )), where P (RN ) isequipped with the weak topology. We choose a test function ϕ ∈ D(RN ). Ifwe neglect all smoothness and existence issues, we can simply write

d

dt

∫ϕ(x)ρ(t, x)dx =

d

dt

∫ϕ(X(t, x))ρ0(x)dx

=∫∇ϕ(X(t, x))

d

dt(X(t, x))ρ0(x)dx

=∫∇ϕ(x)v(t, x)ρ(t, x)dx.

Therefore, ρ(t, ·) satisfies the continuity equation in the distributional sense.

Step II:

We now show that the Wasserstein distance is a lower bound for our func-tional. To do so, we can write

T

∫RN

∫ T

0ρ(t, x)|v(t, x)|2dxdt = T

∫RN

∫ T

0ρ0(x)|v(t,X(t, x))|2dxdt

= T

∫RN

∫ T

0ρ0(x)|

d

dtX(t, x)|2dxdt

Using Jensen’s equation, this leads to

T

∫RN

∫ T

0ρ0(x)|

d

dtX(t, x)|2dxdt ≥

∫RN

ρ0(x)|X(T, x)−X(0, x)|2dx

=∫

RN

ρ0(x)|X(T, x)− x)|2dx

≥ W 22 (ρ0, ρT ).

The last inequality is due to the fact that ∇ψ as well as X(t, x) fullfillcondition 2.3 and ∇ψ is the optimal map. This already shows that the newformulation of the distance is bounded below by the original Wassersteindistance.

2.5. Kantorovich-Wasserstein distances 17

Step III:

Next, we see that the optimal choise of X(t, x) is

X(t, x) = x+t

T(∇ϕ(x)− x) , (2.29)

which impies that there actually exist a combination of (ρ, v) that achievethe infimum, defined by∫

ϕ(x)ρ(t, x)dxdt =∫ϕ(x+

t

T(∇ϕ(x)− x))ρ0(x)dxdt, (2.30)

Assuming that X(t, ·) is invertable we define the velocity field by

v(t, x) =(d

dtX(t, ·)

)X(t, ·)−1 = (T − Id) X(t, ·)−1. (2.31)

We furthermore define v = 0 whenever ρ is zero. As in step I, one can showthat (ρt, vt) also solve the continuity equation. Now choose a nonnegative,measurable function Φ, we have∫

ρ(t, x)Φ(v(t, x))dx =∫ρ0(x)Φ(T (x)− x)dx. (2.32)

Choosing Φ(v) = |v|2, this leads to∫ρt|vt|2dx =

∫ρ0(x)|T (x)− x|2dx. (2.33)

This ensures the existiance of a minizing pair (ρ, v) and therefore concludesthe proof.

Formally, the optimality conditions of this minimization problem can beobtained using the methods of Lagrange multipliers. Here, they turn out tobe

v(t, x) = ∇φ(t, x), (2.34)

where φ is the Lagrange multiplier of the constraints (2.25) and (2.26), andφ satisfies the Hamilton-Jacobi equation

φt = −12|∇φ|2. (2.35)

Remark i) In terms of fluid mechanics, this corresponds to a pressure-


less potential flow.

ii) Considering a metric space endowed with the L2 Kantorovich-Wassersteindistance, the optimality conditions describe, by definition, the geodesicsin this space.

2.6 Displacement Convexity

In this section, we will introduce the concept of diplacement convexity whichwas introduced by McCann in his PhD thesis and in [McCann97] in 1997. Inour discussion, we will mostly follow the presentation in [Villani03, Chapter5]. Futhermore, we will again consider only the quadratic cost function

c(x, y) = |x− y|2.

We start with the following definition.

Definition Let µ, ν be probability measures on RN and assume that theydo not give mass to small sets. Then, according to Brenier’s theorem, thereexists a gradient of a convex function ∇ϕ such that ∇ϕ#µ = ν. Define thedisplacement interpolation as the family of probability measures

ρt := [µ, ν]t = [(1− t)Id + t∇ϕ]#µ. (2.36)

We will now use this interpolation between the probability measures µ and νto define a new form of convexity for functionals on the space of probabilitymeasures.

Definition (Displacement Convexity)

i) A subset P of Pac is called displacement convex if for all µ, ν in P andfor all t ∈ [0, 1], the displacement interpolant ρt is still in P.

ii) If E is a functional on the space P, then E is diplacement convex, if,whenever µ and ν are given elements of P and ρt is the displacementinterpolation between them, t 7→ E(ρt) is convex on [0, 1]. F is said tobe strictly displacement convex, if t 7→ E(ρt) is strictly convex.

2.6. Displacement Convexity 19

iii) A functional is said to be uniformly displacement convex if, for someα > 0,

d2

dt2E(ρt) ≥ αW2(ρ0, ρ1), t ∈ (0, 1). (2.37)

It is said to be semi-displacement-convex, if, for some λ ≥ 0

d2

dt2E(ρt) ≥ −λW2(ρ0, ρ1), t ∈ (0, 1). (2.38)

The concept of displacement convexity will be usefull later interpretated asconvexity along geodesics in a space more complicated that (P,W2).

Chapter 3

Gradient Flow Formulations

of PDEs

3.1 Introduction to Gradient Flows

To introduce gradient flows, we start with an example in the Euclidian space.We fix some entropy functional E ∈ C2(RN ) und study the solution of theordinary differential equation

dx(t)dt

= −gradE(x(t)), x ∈ RN , t ∈ [0,∞). (3.1)

This equation is called steepest descent or gradient flow on the entropy(energy) landscape created by E. We will now investigate proporties ofsolutions of this gradient flow under the assumption that E is uniformlyconvex, following [Carillo06].

Proposition 3.1.1 (Contraction / expansion bounds in a semi-convex val-ley) Fix k ∈ R. If E ∈ C2(RN ) is uniformly convex, i.e. D2E(X) ≥ kId forall x ∈ RN , and the curve x(t), y(t) ∈ RN , t ∈ [0,∞) both solve (3.1), then

|x(t+ t0)− y(t+ t0)| ≤ e−kt|x(t0)− y(t0)|. (3.2)

Proof To proof this, we first define the linear interpolation between x(t)and y(t) by

g(s) = (1− s)x(t) + ty(t). (3.3)

21

22 Chapter 3. Gradient Flow Formulations of PDEs

With f(t) = |x(t)− y(t)|2/2 it follows

f ′(t) = −〈x(t)− y(t),∇E(x(t))−∇E(y(t))〉

= −⟨x(t)− y(t),

∫ 1

0

d

ds(∇E(g(s))) ds

⟩= −

⟨x(t)− y(t),

∫ 1

0D2E(g(s))

dg(s)ds

ds

⟩≤ −

⟨x(t)− y(t),

∫ 1

0kId(x(t)− y(t))ds

⟩= −2kf(t).

Using Gronwalls inequality [Gronwall19], this leads to f(t+t0) ≤ e−2ktf(t0).

If k > 0, the above proposition shows that the solution of the map

x(0) ∈ RN → Xt(x(0)) = x(t)

of the initial value problem of (3.1) defines a uniform contraction on RN foreach t ≤ 0. The fact that this map is well-defined follows locally in space andtime from the C2-smoothness of E. To show that it is globally well-definedfor all future times, we make use of the fact that E is coercive which followsfrom its uniform convexity and smoothness. Using Taylors formula we get

E(x) ≥ E(x(0))+ < ∇E(x(0)), x− x(0) > +k|x− x(0)|2

2. (3.4)

This proporty ensures the compactness of the level sets

SE(x(0)) = x|E(x) ≤ E(x(0)) , (3.5)

which contain x(t) for all t.Furthermore, since RN is complete, from the contraction mapping principlefollows the existance of a unique fixpoint x∞ of the map Xt as well as thefact that every solution of (3.1) converges to x∞ for t→∞.

3.2. Otto’s Approach for the porous medium equation 23

3.2 Otto’s Approach for the porous medium equa-

tion

In his by now famous paper [Otto01], Otto formally derives a gradient flowformulation for the porous medium equation

ρt −∆(ρm) = 0, (3.6)

where ρ ≥ 0 is a density function on RN , t ∈ [0,∞) is the time and x ∈ RN .Furthermore, ∆ is the laplace operator with respect to the spatial variablex.

Remark i) For m = 1 this is the heat equation. For m > 1, the equa-tion is called slow diffusion equation or for 0 < m < 1 fast diffusionequation respectivly.

ii) In the following, for technical reasons, we restrict the values of m via

m >N

N + 2and m ≥ 1− 1

N. (3.7)

As a consequence of the gradient flow interpretation of this equation, Otto(formally) maintains convergence rates to the asymptotic solutions of theporous medium equation by standard Riemannian calculus.

Remark In case of the porous medium equation, the stationary solutionsare the so-called Prattle-Barenblatt [Pattle59] solutions. These are self-similar solutions of the form

ρ∞(t, x) =1tNα

ρ∞

( xtα

), (3.8)

where the function ρ∞ is implicitly given by

e′(ρ∞(y)) =

mm−1 ρ∞(y)m−1 = maxλ− α1

2 |y|2, 0 for m > 1

ln ρ∞(y) + 1 = λ− α12 |y|

2 for m = 1mm−1 ρ∞(y)m−1 = λ− α1

2 |y|2 for m < 1

.

(3.9)Furthermore, α is given by

α =1

N(m− 1) + 2(3.10)


and λ is chosen such that ∫ρ∞(y)dy = 1. (3.11)

The Bahrenplatt-Prattle profiles are stationary solutions of the porous mediumequation in the sense that, rescaling time and space acccording to

x = tαy t = exp(τ),

the rescaled solution (which we will call ρ in the following) approaches theBahrenblatt-Prattle profiles for large times.

3.2.1 Gradient Flow Formuation

The setting in which the gradient flow will be established is the following.First, a differentiable manifold M is defined by

M =

non-negative functions ρ on RN with∫ρ = 1

. (3.12)

Furthermore, the tangent space is given by

TρM =

functions s on RN with∫s = 0

. (3.13)

To define a metric tensor, we use the following identification of the trangentspace.

TρM =functions p on RN

/ ∼ (3.14)

where the identification between p and s is via

−div(ρ∇p) = s. (3.15)

The ∼ means that p’s, which differ only by a constant are indentified. Then,the metric tensor is defined as

gρ(s1, s2) =∫ρ∇p1∇p2. (3.16)

3.2. Otto’s Approach for the porous medium equation 25

We define as a gradient flow on this manifold M the dynamical system givenby

dρ

dt= −gradE |ρ (3.17)

Now, we want to represent the porous medium equation as a gradient flowon this manifold. Therefore, we define the Energy functional

E(ρ) =

1

m−1

∫ρm for m 6= 1∫

ρ ln(ρ) for m = 1

. (3.18)

Thus the differential is given by

diffE(ρ) · s =

∫mm−1ρ

m−1s for m 6= 1∫(ln(ρ) + 1)s for m = 1

(3.19)

Using the definition of the gradient flow on M, we get ∫ρtp+

∫mm−1ρ

m−1s = 0 for m 6= 1∫ρtp+

∫(ln(ρ) + 1)s = 0 for m = 1

(3.20)

using equation (3.15), this leads to ∫ρtp+

∫mm−1ρ

m−1∇ · (ρ∇p) = 0 for m 6= 1∫ρtp+

∫(ln(ρ) + 1)s∇ · (ρ∇p) = 0 for m = 1

(3.21)

and finally, integrating by parts∫(ρt −∆ρm)p = 0. (3.22)

As p is arbitrary, we recover the porous medium equation.This calculation shows that the porous mediums equation can indeed beunderstood as a gradient flow on a metric space M endowed with the metrictensor as defined in 3.16.

3.2.2 Formal Derivation of Convergence Rates to Stationary

Solutions

In [Otto01, Section 3.2] Otto states three inequalities that show convergenceto the stationary state, which he later proofs by formal Riemannian calculus.To do so, we assume the following conditions:


i) ρ evolves according to the gradient flow

dρ

dτ= gradF |ρ (3.23)

on (M, g), where the augmented functional F is given by

F (ρ) = E(ρ) + α

∫12|y|2ρ(y)dy. (3.24)

ii) ρ∗ is a minimizer of F on M, i.e.

F (ρ) ≥ F (ρ∗) ∀ρ ∈M. (3.25)

This implies also that−gradF |ρ∗ = 0. (3.26)

iii) F is uniformly strictly (wirklich) convex on (M, g), i.e.

HessF |ρ ≤ αId ∀ρ ∈M, (3.27)

in the sense of

⟨s,Hess(F |ρ)s

⟩≥ α|s|2 ∀s ∈ TρM, ∀ρ ∈M. (3.28)

This follows from

HessE |ρ ≥ 0 and HessM |ρ = Id ∀ρ ∈M. (3.29)

Note that the ealier condition m ≥ 1− 1N ensures the convexity of E

whereas the condition m ≥ NN+2 ensures that both E(ρ∗) and M(ρ∗)

are well-defined and finite.

Using these assumptions, Otto derives

d

dτ

(exp(2ατ)|gradF |ρ|2

)≤ 0, (3.30)

d

dτ(exp(2ατ)(F (ρ)− F (ρ∞))) ≤ 0, (3.31)

d

dτ

(exp(2ατ)d(ρ, ρ∞)2

)≤ 0. (3.32)

3.3. Gradient flow for Systems 27

All these equation mean that the rate of convergence towards ρ∞ is exponen-tial with a rate α with respect to τ and with a polynomial rate with respectto t. In the first equation, the distance between the stationary solution andthe solution and ρ is measured. In the second equation, the distance of ρfrom a minimizer of F is measured and in the third the distance between ρand ρ∗.The important fact is, that Otto was able to derive all these equations byrelatively simple calculus on the Riemannian manifold M . We illustrate thisfor the first equation, which is the easiest, only.

d

dτ|gradF |ρ|2 = 2

⟨gradF |ρ,

D

DτgradF |ρ

⟩= −2

⟨gradF |ρ,HessF |ρgradF |ρ

⟩≤ −2α|gradF |ρ|2.

Using some sort of Gronwall-type inequality, this directly leads to 3.30.

3.3 Gradient flow for Systems

In this chapter, we try to generalize Otto’s ideas to systems of diffusiveequations. Therefore, we first show that the equation (or the system ofequations) we consider can also be seen as a gradient flow. However, weneed to define a more complicated metric tensor on this manifold. Then,we try to use displacement convexity to prove convergence to stationarysolutions.

3.3.1 Introduction

In the following, we will use the methods described above to analyse theequation

ρt = div(D(ρ)(∇u(ρ) +∇V (x)), i = 1, . . . , n x ∈ RN , t > 0 (3.33)

with initial condition ρ(0) = ρ0.We will distinguish between the scalar and and the vector valued case. Inthe scalar case, ρ(x, t) ∈ R and D(ρ) is a real valued function of ρ. In thevector valued case, ρ = (ρ0, . . . , ρn) ∈ RN and D is a matrix-valued functionof ρ. In this case D is assumed to be symmetric and positive definite. The


functions u and ρ are linked by the algebraic relation ρ = ρ(u). u and thepotential V (x) are also either scalar or vector-valued.

Examples

Setting n = 1, D(ρ) = ρ, u(ρ) = log ρ, the system turns into the heatequation

ρt = ∆ρ. (3.34)

For n = 2, ρ = (ρ1, ρ2) and u = (u1, u2) are related by

ρ1 = (−u2)−3/2eu1 ρ2 = −32ρ1

u2(3.35)

the system turns into the energy-transport-equations.

The scalar case and the Otto formulation

We show that in the scalar case without potential, equation (3.33) is equiv-alent to the porous medium equation in Otto’s formulation.

ρt = div(∇f(ρ)) = ∇ · (f ′(ρ)∇ρ). (3.36)

For the system formulation

ρt = div(D(ρ)∇h′(ρ)) = ∇ · (D(ρ)h′′(ρ)∇ρ), u = h′ (3.37)

holds. Settingf ′(ρ) = D(ρ)u′′(ρ) (3.38)

the two formulations are equal for the scalar case.

Remark Assuming D to be positive, (3.38) means that monotonicity of f(f ′ > 0) and convexity of u (u′′ > 0) are equivalent.

3.3.2 Gradient flow interpretation

In this section, we show that in the vector valued case without a potential,the equation can still be considered as a gradient flow.Let

M =ρi ≥ 0, i = 1 . . . N |

∫Rd

ρi dx = 1

(3.39)


be a manifold and

TρM =ρi ≥ 0, i = 1 . . . N |

∫Rd

ρi dx = 0

(3.40)

be the tangent space at ρ. For a given ρ define the metric tensor

gρ(p, q) =∫

Rd

∇vTD(ρ)∇u dx, p, q ∈ TρM (3.41)

where ui and vj are the solutions of

−div(D(ρ)∇u) = p, −div(D(ρ)∇v) = q,

p = (p1, . . . , pn), q = (q1, . . . , qn), etc.We assume that the functional setting is such that the above elliptic systemsare uniquely solvable. Note that the metric tensor can be rewritten as

gρ(p, q) =∫

Rd

p · vdx =∫

Rd

q · u dx (3.42)

In the following lemma we prove that, using these definitions, (3.33) can infact be interpreted as a gradient flow on M .

Lemma 3.3.1 Assume that the setting is such that the formula gρ(gradE, p) =dE · p for all vector fields p on M holds. Furthermore, suppose that thereexists a function h : Rn → R such that u(ρ) = h′(ρ). Then (3.33) can bewritten as

ρt = −gradE|ρ (3.43)

where the entropy E is given by

E(ρ) =∫

Rd

h(ρ) dx (3.44)

Proof This is a slight generalization of Otto’s argument [Otto01]. Letp = (p1, . . . , pn) ∈ TρM be a vector field and v be a vector-valued func-tion defined by p = −div(D(ρ)∇v).


Then

gρ(ρt, p) + dE · p =∫

Rd

[ρt · v + h′(ρ) · p

]dx

=∫

Rd

[ρt · v − h′(ρ)∇ · (D(ρ)∇v)

]dx

=∫

Rd

[ρt · v + (∇h′(ρ)) : (D(ρ)∇v)

]dx

=∫

Rd

[ρt · v + (D(ρ)∇h′(ρ)) : ∇v

]dx

=∫

Rd

[ρt · v −∇ · (D(ρ)∇h′(ρ))v

]dx

=∫

Rd

[ρt −∇ · (D(ρ)∇u(ρ)︸︷︷︸=0, (5.15)

]v dx

= 0 ∀v ∈ Rd

Thus, gρ(ρt, p) = −gρ(gradE|ρ , p) for all p and the conclusion follows.

Remark i) This calculation includes the scalar case for d = 1.

ii) Adding a potential, this calculation still works

gρ(ρt, p) + dE · p =∫

Rd

[ρt · v + (u′(ρ) + V ) · p

]dx

=∫

Rd

[ρt · v − (u′(ρ) + V )div(D(ρ)∇v)

]dx

=∫

Rd

[ρt · v +∇(h′(ρ) + V ) : (D(ρ)∇v)

]dx

=∫

Rd

[ρt · v +D(ρ)∇(h′(ρ) + V ) : ∇v

]dx

=∫

Rd

[ρt · v − div(D(ρ)∇(h′(ρ) + V ))v

]dx

=∫

Rd

[ρt − div(D(ρ) (∇u(ρ) +∇V ))︸︷︷︸=0, (5.15)

]v dx

= 0 ∀v ∈ Rd


A consequence of the above lemma is that the entropy is decreasing alongthe trajectories

dE

dt= dE|ρ · ρt = gρ(gradE, ρt) = −gρ(ρt, ρt) ≤ 0. (3.45)

Examples

Let n = 1, Dij(ρ) = ρ, and u(ρ) = αα−1ρ

α−1, where α > 0 (this is the slow(α > 1 or fast α ≤ 1 diffusion case). Then h(ρ) = 1

α−1ρα and we are in the

setting of [Otto01].

3.3.3 Distance and Geodesics on (M, g)

To get a distance on the Space (M, g), we generalize the idea of Benamouand Brenier (2.27) and define

d2ρ(ρ0, ρT ) =

12

inf(ρ,v)

∫ T

0

∫RN

tr(vTD(ρ)v) dxdt (3.46)

under the constraints

div(D(ρ)v) = ρt, ρ(0) = ρ0, ρ(T ) = ρT (3.47)

Again, we use the method of Lagrange multipliers to formally derive theoptimality conditions for this minimization problem.

Lemma 3.3.2 The formal solution of the constrained minimization problemstated above is given by v = −∇λ and

λt =12

∑k,l,m

∇ρDkl(ρ)vkmvlm, λ(0) = −µ0, λ(T ) = µT , (3.48)

div(D(ρ)v) = ρt, ρ(0) = ρ0, ρ(T ) = ρT . (3.49)

Proof We start with the minimization problem. The Lagrangian is by


definition given as

L =12

∫ T

0

∫RN

tr(vTD(ρ)v) + λ · ρt − λ · div(D(ρ)v) dxdt

+∫

RN

µ0(ρ(0)− ρ0) + µT (ρ(T )− ρT ) dx

=12

∫ T

0

∫RN

∑k,l,m

Dkl(ρ)vkmvlm + λ · ρt − λ · div(D(ρ)v) dxdt

+∫

RN

µ0(ρ(0)− ρ0) + µT (ρ(T )− ρT ) dx

=: Lλ + Lµ.

where λ ∈ RN and µ = (µ0, µT ) are the Lagrange multiplier. We firstcalculate the derivative of Lλ with respect to v into a arbitrary direction w.

∂Lλ∂v

· w =∫ T

0

∫RN

∑k,l,m

Dkl(ρ)vkmwlm +∑k,l,m

Dkl(ρ)wlm∂λk∂xm

dxdt

=∫ T

0

∫RN

∑k,l,m

Dkl(ρ)(vkm +∂λk∂xm

)wlm dxdt ∀w.

From this followsv = −∇λ. (3.50)

Next, the derivative with respect to ρ into direction g is given by

∂Lλ∂ρ

· g =12

∫ T

0

∫RN

∑k,l,m

(∇ρDkl(ρ) · g)vkmvlm − λt · g

−∑k,l,m

(∇ρDkl(ρ) · g)vkm∂λl∂xm

dxdt

=∫ T

0

∫RN

−12

∑k,l,m

(∇ρDkl(ρ) · g)vkmvlm − λt · g dxdt ∀g.

From this follows, after partial integration with respect to time:

λt =12

∑k,l,m

∇ρDkl(ρ)vkmvlm. (3.51)

For the constraint, we consider the derivative of Lµ. As the derivative with


respect to v is zero, we only look at(∂Lλ∂ρ

+∂Lµ∂ρ

)· g =

∫ T

0

∫RN

−12

∑k,l,m

(∇ρDkl(ρ) · g)vkmvlm − λt · g dxdt

+∫

RN

(µ0 · g(0) + µT · g(t)) dx.

Partial integration with respect to time and (3.51) finally reads to

λ(0) = −µ0, λ(T ) = µT . (3.52)

Remark In contrast to Benamou and Brenier, in this case the two equations(3.48) and (3.49) are coupled.

Chapter 4

Entropy Dissipation Methods

In this chapter, so-called entropy methods for PDEs will be discussed. Es-pecially, their use to obtain convercence rates towords stationary sulutionswill be regarded. We will mainly follow the discussion in [Carillo01]. For amore general review on the connection between PDEs and entropy as wellas for the physical meaning of the entropy, see [Evans04].The strategy will be the following. After defining an entropy functional,one shows that this functional is decreasing in time. Furthermore, it can beshown that the asymptotic solution is a minimizer of the entropy functional.Theorefore, one uses the relative entropy to measure the distance betweena sulution at time t and the stationary solution. Finally, one uses a Csiszar-Kullback [Csisz’ar67, Kullback59] type inequality to proof a convergencerate in L1.

4.1 Principle

To demonstrate the method, we consider the scalar, nonlinear Fokker-Planckequation

ut = div(u∇V (x) +∇f(u)), x ∈ RN , t > 0, (4.1)

u(x, t = 0) = u0(x) ≥ 0, x ∈ RN . (4.2)

We assume that f and V are such that there exists a unique stationarysolution u∞. Furthermore, we assume f ′ and u to be positive, so that the

35

36 Chapter 4. Entropy Dissipation Methods

equation is parabolic. Note that the equation is mass preserving, i.e.∫u(x, t)dx =

∫u0(x)dx =: M. (4.3)

This is due to the fact that it is in divergence form. In the following, weonly consider the simple case f(u) = u.

Entropy

It can be checked that for every convex function φ in R, the equation

E(u(t)) =∫φ(u(t))dx (4.4)

defines a Lyaponov functional along the solutions of the PDE, i.e. its der-vivative with respect to t is the negative of a positive functional. Here, wedefine the entropy making the choice

φ(x) := u(log(u)− 1) + 1. (4.5)

Then, the entropy itself is defined by

E(u(t)) =∫φ(u(x, t)) + u(x, t)V (x)dx. (4.6)

Assuming V , u and φ(u) to be in L1(RN ), we can compute the first derivativein time.

dE(u(t))dt

=∫ut(log(u) + V (x))dx

=∫

div(u∇V (x) +∇u)(log(u) + V (x))dx

= −∫ [

u∇V (x) +∇u(∇uu

)+∇V (x)

]dx

= −∫u(x, t)

∣∣∣∣∇V (x) +∇uu

∣∣∣∣2 dxThis gives an expression for the dissipation of the entropy and

I(u(t)) =∫u(x, t)

∣∣∣∣∇V (x) +∇uu

∣∣∣∣2 dx (4.7)

4.1. Principle 37

is called entropy production functional. This implies that the Entropy is aLyaponov functional (along solutions) since u ≥ 0. The next step in thestategy outlined above is to differentiate the entropy production functional.If the resulting expression is of the form such that for λ ≤ 0,

dI(u(t))dt

= −λI(u(t))−R(t), (4.8)

with some reminder R(t) ≥ 0 on RN , using (4.7), we imediately get

I(u(t)) =dE(u(t))

dt=

1λ

[dI(u(t))

dt+R(t)

]. (4.9)

Using the fact that R(t) is positive, this leads to

dI(u(t))dt

≤ λI(u(t)). (4.10)

This implies (using Gronwall’s inequality)

I(u(t)) ≤ e−λtI(u(0)) (4.11)

Because the stationary solution of is a minimizer of the entropy functional,the value of the entropy production functional at the stationary solution iszero. Therefore, integrating the previous expression from zero to infinityand using the fact that R(t) is positive,

E(u0)− E(u∞) ≤ 1λI(u0) (4.12)

and.d[E(u(t))− E(u∞)]

dt≤ −λ[E(u(t))− E(u∞)] (4.13)

hold. Introducing the relative entropy E(u(t)|u∞) by

E(u(t)|u∞) = E(u(t))− E(u∞), (4.14)

the last equation can be written as

E(u(t)|u∞) ≤ I(u(t)|u∞). (4.15)

38 Chapter 4. Entropy Dissipation Methods

This equation is called log-Sobolev inequality. This is exponential conver-gence of the entropy functional towards its minumum at a rate λ, usingGronwalls lemma [Gronwall19] Using a Csiszar-Kullback type inquality, thefinal result is convergence in L1 at a rate of λ/2.

Chapter 5

Application to Systems of

Diffusive PDEs

In this Chapter both the entropy dissipation and the gradient flow methodwill be used to examine the long-time behaviour of nonlinear Fokker-Planckequations of the type

ρt = div(D(ρ)(∇u′(ρ) +∇V

)). (5.1)

There ρ = ρ(x, t), the potential V depends only on x and ′ denotes thederivative with respect to ρ. First, the case of a scalar non-linearity D(ρ)will be treated. This leads to different results for both methods. To explainthese differences, both methods will be appled to the fully one-dimensionalcase.Finally, system case will be treated but only for the example of energytransport.

5.1 Entropy Dissipation Methods

To use the entropy dissipation method we first define the entropy functionalas

E(ρ) =∫u(ρ) + ρV (x)dx. (5.2)

We will examine the derivatives of this functional along the solutions of thePDE.

39

40 Chapter 5. Application to Systems of Diffusive PDEs

5.1.1 The Scalar Case

We start showing that the first derivative of E (along solutions of (5.15)) isthe negative of some entropy production functional.

dE

dt=

∫ρt(u′(ρ) + V (x)) dx

= −∫D(ρ)|∇u′(ρ) +∇V (x)|2 dx

= −∫D(ρ)|ξ|2 dx,

withξ = ∇u′(ρ) +∇V.

Next, we study the derivative of this production functional (i.e. secondderivative of the entropy).

dI

dt(u(t)) = −d

2E

dt2=∫ρtD

′(ρ)|ξ|2dx+ 2∫D(ρ)ξ · ∂ξ

∂tdx =: I1 + I2. (5.3)

For the second term we get, after partial integration,

I2 = −2∫

div(D(ρ)ξ)∂u′

∂t(ρ)dx = −2

∫u′′(ρ)(div(D(ρ)ξ))2dx. (5.4)

For the first term we get

I1 =∫

div(D(ρ)ξ)D′(ρ)|ξ|2dx = −∫D(ρ)ξ · ∇

(D′(ρ)|ξ|2

)dx

= −∫D(ρ)D′′(ρ)|ξ|2(ξ · ∇ρ)dx− 2

∫D(ρ)D′(ρ)ξTJacob(ξ)ξdx

=: I(1)1 + I

(2)1 .

The second term can we written as

I(2)1 = −2

∫D(ρ)D′(ρ)ξTHess(V )ξdx− 2

∫D(ρ)D′(ρ)ξTHess(u′(ρ))ξdx.

(5.5)Our goal now is to apply Bochner’s formula (see below) and therefore wewill have to rewrite the second part of (5.5). Assuming D(ρ) 6= 0, we cando the following calculation.

5.1. Entropy Dissipation Methods 41

− 2∫D(ρ)D′(ρ)ξTHess(u′(ρ))ξdx

= −2N∑

i,j=1

∫D′(ρ)(D(ρ))−1(D(ρ)ξi)(D(ρ)ξj)

∂2u′(ρ)∂xi∂xj

= −2N∑

i,j=1

∫∂u′(ρ)∂xi

(D′(ρ))2(D(ρ))−2 ∂ρ

∂xj(D(ρ)ξi)(D(ρ)ξj)

+ 2N∑

i,j=1

∫∂u′(ρ)∂xi

D′(ρ)(D(ρ))−1∂ [(D(ρ)ξi)(D(ρ)ξj)]∂xj

+ 2N∑

i,j=1

∫∂u′(ρ)∂xi

∂ρ

∂xjD′′(ρ)(D(ρ))−1(D(ρ)ξi)(D(ρ)ξj)︸︷︷︸

=:I(3)1

= I(3)1 − 2

N∑i,j=1

∫(D′(ρ))2u′′(ρ)

∂ρ

∂xi

∂ρ

∂xjξiξj

+ 2N∑

i,j=1

∫∂u′(ρ)∂xi

D′(ρ)(ξi∂(D(ρ)ξj)

∂xj+ ξj

∂(D(ρ)ξi)∂xj

)

= I(3)1 − 2

N∑i,j=1

∫(D′(ρ))2u′′(ρ)

∂ρ

∂xi

∂ρ

∂xjξiξj

+ 4N∑

i,j=1

∫∂u′(ρ)∂xi

D′(ρ)ξi∂(D(ρ)ξj)

∂xj

− 2N∑

i,j=1

∫∂u′(ρ)∂xi

D′(ρ)(ξi∂(D(ρ)ξj)

∂xj− ξj

∂(D(ρ)ξi)∂xj

)

= −2N∑

i,j=1

∫(D′(ρ))2u′′(ρ)

∂ρ

∂xi

∂ρ

∂xjξiξj

+ 4N∑

i,j=1

∫∂u′(ρ)∂xi

D′(ρ)ξi∂(D(ρ)ξj)

∂xj

− 2∫D(ρ)D′(ρ)u′′(ρ)

[(ξ · ∇ρ) div(ξ)− 1

2∇(|ξ|2) · ∇ρ

]dx+ I

(3)1


We now collect all the above terms. Furthermore, we define f via f ′ =D(ρ)u′′(ρ). Hence,

I1 + I2 = I(1)1 + I

(3)1 − 2


∫D(ρ)f ′(ρ)(div(ξ))2dx

− 2∫D′(ρ)f ′(ρ)

[(ξ · ∇ρ) div(ξ)− 1

2∇(|ξ|2) · ∇ρ

]dx.

This can be rewritten as

I1 + I2 = I(1)1 + I

(3)1 − 2


∫D(ρ)f ′(ρ)(div(ξ))2dx

− 2∫D′(ρ)

[(ξ · ∇f(ρ)) div(ξ)− 1

2∇(|ξ|2) · ∇f(ρ)

]dx.

Then, partial integration leads to

I1 + I2 = I(1)1 + I

(3)1 − 2

∫D(ρ)D′(ρ)ξTHess(V )ξdx

− 2∫

(D(ρ)f ′(ρ)−D′(ρ)f(ρ))(div(ξ))2dx

− 2∫D′(ρ)f(ρ)

[12∆(|ξ|2)− (ξ · ∇div(ξ))

]dx

− 2∫D′′(ρ)f(ρ)

[12∇(|ξ|2) · ∇ρ− (ξdiv(ξ) · ∇ρ)

]dx.

Now, we can really use Bochner’s formula (for a space with Ricci curvaturezero),

12∆(|ξ|2)− (ξ · ∇div(ξ)) = tr(∇ξ)2, (5.6)

which results in

I1 + I2 = −2∫D(ρ)D′(ρ)ξTHess(V )ξdx−

∫D(ρ)D′′(ρ)|ξ|2(ξ · ∇ρ)dx

− 2∫

(D(ρ)f ′(ρ)−D′(ρ)f(ρ))(div(ξ))2dx− 2∫D′(ρ)f(ρ)tr(∇ξ)2dx

− 2∫D′′(ρ)f(ρ)

[12∇(|ξ|2) · ∇ρ− (div(ξ)ξ · ∇ρ)

]dx

+ 2∫D(ρ)D′′(ρ)u′′(ρ)|ξ|2|∇ρ|2dx.

5.1. Entropy Dissipation Methods 43

Furthermore


∫D(ρ)D′′(ρ)|ξ|2(ξ · ∇ρ)dx

− 2∫


− 2∫D′′(ρ)f(ρ)(∇u′ +∇V )T (Hess(

du

dρ(ρ)) + Hess(V ))∇ρdx

+ 2∫D′′(ρ)f(ρ)div(ξ)

(∇u′ · ∇ρ+∇V · ∇ρ

)dx

+ 2∫D(ρ)D′′(ρ)u′′(ρ)|ξ|2|∇ρ|2dx,

and


∫D(ρ)D′′(ρ)|ξ|2(ξ · ∇ρ)dx

− 2∫


− 2∫D′′(ρ)f(ρ)(u′′∇ρTHess(u′(ρ))∇ρ+ u′′∇ρTHess(V )∇ρ)dx

− 2∫D′′(ρ)f(ρ)(∇ρTHess(u′(ρ))∇V +∇ρTHess(V )∇V )dx

+ 2∫D′′(ρ)f(ρ)div(ξ)

(u′′|∇ρ|2 +∇V · ∇ρ

)dx

+ 2∫D(ρ)D′′(ρ)u′′(ρ)|ξ|2|∇ρ|2dx.

Setting V = 0, which implies ξ = ∇u′ we get

I1 + I2 = −2∫

(D(ρ)f ′(ρ)−D′(ρ)f(ρ))(div(∇u′))2dx− 2∫D′(ρ)f(ρ)tr(∇2u′)2dx

− 2∫D′′(ρ)f(ρ)u′′(ρ)(∇ρTHess(u′(ρ))∇ρ)dx

+ 2∫D′′(ρ)f(ρ)div(∇u′)u′′|∇ρ|2dx

+∫D(ρ)D′′(ρ)u′′(ρ)|ξ|2|∇ρ|2dx.


5.2 Gradient Flow and Displacement Convexity

We will now use the method of displacement convexity of examine theasymptotic behaviour of solutions of (5.15).

5.2.1 Linear Scalar Case

First, we consider the case of a scalar ρ ∈ R and D(ρ) = ρ. This means,we recover the porous medium equation. Following [Otto05], we write theequation in the form

ρt = div(∇u′(ρ) +∇V ). (5.7)

and define the Entropy by

E(ρ) =∫u(ρ) + ρV (x)dx =: Eint + Epot. (5.8)

First, we take care of the internal energy. Recall that displacement convexitymeans convexity along geodesics which are parametrised by s here. Firstderivative is

dE

ds=∫u′(ρ)ρs = −

∫u′′∇ρ · ρv = −

∫∇f(ρ) · v, (5.9)

when f is defined viaf ′(ρ) = ρu′′(ρ).

The Second derivative turns out to be

d2E

ds2=

∫RN

f ′(ρ)div(ρv)div(v) + f(ρ)div(vs)dx

=∫

RN

ρf ′(ρ)(div(v))2 +∇f(ρ)vdiv(v) + f(ρ)∆(

12|v|2)dx

=∫

RN

ρf ′(ρ)(div(v))2 + f(ρ)−div(vdiv(v)) + ∆

(12|v|2)

dx

=∫

RN

ρf ′(ρ)(div(v))2 + f(ρ)−(div(v))2 − v · ∇div(v) + ∆

(12|v|2)

dx

=∫

RN

(ρf ′(ρ)− f(ρ)

)(div(v))2 + f(ρ)

−v · ∇div(v) + ∆

(12|v|2)

dx

=∫

RN

(ρf ′(ρ)− f(ρ)

)(div(v))2 + f(ρ)tr(Dv)2dx.

5.2. Gradient Flow and Displacement Convexity 45

In this calculation we used

div(vdiv(v)) = (div(v))2 + v · ∇div(v) (5.10)

and (again) Bochner’s formula for a space with Ricci curvature zero

∆12|v|2 − v · ∇div(v) = tr(Dv)2. (5.11)

Obviously, the second derivative is positive for ρ ≥ 0 and

ρf ′(ρ)−(

1− 1n

)f(ρ) ≥ 0, (5.12)

because of div(v)2 ≤ ntr(Dv)2.

Example

As a simple example we show that for the choice

u′(ρ) = log(ρ),

the above condition is always fulfilled. We get

f ′(ρ) = 1 ⇒ f(ρ) = ρ.

Inserting this into (5.12) leads to the new condition

1nρ ≥ 0,

which is always fulfilled for non-negative ρ.

Potential Energy

Adding a Potential to the equation does not alter the situation as long asthe potential is convex. This is shown by the following calculation.The part of the entropy corresponding to the potential is given by

E(ρ) =∫ρV (x) dx. (5.13)


The derivatives of this entropy functional are

∂E

∂s= −

∫ρv∇V dx (5.14)

and

∂2E

∂s2= −

∫ρsv∇V dx−

∫ρvs∇V dx

= −∫

div(ρv)v∇V dx−∫ρ∇(

12|v|2)∇V dx

=∫ρv∇(v∇V ) dx−

∫ρ∇v · v∇V dx

=∫ρv∇v∇V dx+

∫ρvTHess(V )v dx−

∫ρ∇v · v∇V dx

=∫ρvTHess(V )v dx.

This shows that the sign the second derivative is really given by the convexityproperties of the potential as stated above.

5.2.2 Nonlinear Scalar Case with Potential

We start with the equation in the form

ρt = div(D(ρ)

(∇u′(ρ) +∇V

)), (5.15)

where ρ = ρ(x, t), the potential V depends only on x and ′ denotes thederivative with respect to ρ. The Entropy is defined as

E(ρ) =∫u(ρ) + ρV (x)dx. (5.16)

Again we study the derivatives of the functional E(ρ(s)) along the geodesicswhich are parametrised via the parameter s.

dE

ds=∫

RN

ρsu′(ρ) dx = −

∫RN

D(ρ)v∇u′(ρ) dx = −∫

RN

D(ρ)v∇(ρ)u′′(ρ) dx.

(5.17)Defining f(ρ) via f ′(ρ) = D(ρ)u′′(ρ) we obtain

dE

ds= −

∫RN

∇f(ρ)v dx. (5.18)


For the second derivative we get

d2E

ds2= −

∫RN

∇(ρsf ′(ρ)) · v −∫

RN

∇(f(ρ))vs dx

=∫

RN

div(D(ρ)v)f ′(ρ)div(v) dx− 12

∫RN

∇f(ρ)∇(D′(ρ)|v|2

)dx

=∫

RN

D(ρ)f ′(ρ)(div(v))2 dx+∫

RN

D′(ρ)∇ρf ′(ρ)vdiv(v) dx

− 12

∫RN

∇f(ρ)∇(D′(ρ)

)|v|2 dx−

∫RN

D′(ρ)∇ρf ′(ρ)∇(|v|2)dx

Now defining g via g′(ρ) = D(ρ)D′(ρ)u′′(ρ) we can write

d2E

ds2=

∫RN

D(ρ)f ′(ρ)(div(v))2 dx−∫

RN

g(ρ)div(vdiv(v)) dx

− 12

∫RN

∇f(ρ)∇(D′(ρ)

)|v|2 dx−

∫RN

g(ρ)∆(|v|2)dx.

Usingdiv(vdiv(v)) = (div(v))2 + v · ∇div(v) (5.19)

we get

d2E

ds2=

∫RN

D(ρ)f ′(ρ)− g(ρ)

(div(v))2 dx+

∫RN

12∆(|v|2)− v · ∇div(v)

g(ρ) dx

− 12

∫RN

D(ρ)u′′(ρ)D′′(ρ)|∇ρ|2|v|2dx

and with Bochner’s formula (again for a space with Ricci curvature zero)

∆12|v|2 − v · ∇div(v) = tr(∇v)2. (5.20)

we finally obtain

d2E

ds2=

∫RN

D(ρ)f ′(ρ)− g(ρ)

(div(v))2dx+

∫RN

g(ρ)tr(∇v)2 dx

− 12

∫RN

D(ρ)u′′(ρ)D′′(ρ)|∇ρ|2|v|2 dx.

Because of div(v)2 ≤ ntr(∇v)2, the first two terms are positive if ρ ≥ 0 and

D(ρ)f ′(ρ)−(

1− 1n

)g(ρ) ≥ 0. (5.21)


The second term is only positive, if D′′(ρ) ≤ 0, assuming D(ρ) and u′′(ρ) tobe nonnegative.

Examples

We will examine when the conditions for (5.12) will be fulfilled for threetypical examples.

i) To check if our results are compatible with the linear case, we writethe linear equation in the form

ρt = div(D(ρ)∇u′(ρ)

)with u′(ρ) defined via

u′′(ρ) =1

D(ρ).

This leads tog′(ρ) = D(ρ)D′(ρ)u′′(ρ) = D′(ρ),

and therefore1nD(ρ) ≥ 0.

Again we see that this condition is always fulfilled (as long as D ispositive).

ii) As a second example we choose

D(ρ) = ρα, u′(ρ) = log(ρ), α ≥ 0.

This leads tof ′(ρ) = D(ρ)u′′(ρ) = αρα−1,

and by integrating

f(ρ) =1αρα.

Inserting these formula’s into (5.12), we obtain

ρα(

1− 1α

+1αn

)≥ 0.

As ρ is always nonnegative for nonnegative initial data, this leads to


the conditionα ≥

(1 +

1n

).

iii) For the third example we choose

D(ρ) = ρ(1− ρ), u′(ρ) = log(ρ).

As above we get

g′(ρ) = (1− ρ)(1− 2ρ) ⇒ g(ρ) =23ρ3 − 3

2ρ2 + ρ,

andD(ρ)D′(ρ)

=ρ(1− ρ)1− 2ρ

.

If we put these two results in (5.12), the result is

ρ(1− ρ)1− 2ρ

(1− ρ)(1− 2ρ)−(

1− 1n

)(23ρ3 − 3

2ρ2 + ρ

)≥ 0

⇒ ρ

(ρ2

(13

+23n

)− ρ

(12

+32n

)+

1n

)≥ 0

For large ρ (i.e. for ρ close to 1) this leads to the condition

13

+23n

− 12− 3

2n+

1n

= −16

+4− 9 + 6

6n= −1

6

(1− 1

n

).

This term is negative for every n > 1.

Potential Energy Term

As above, we examine the effect of an additional potential. Again, theentropy is defined as

P (ρ) =∫ρV (x) dx. (5.22)

The first and second derivative are

∂P

∂s= −

∫D(ρv)∇V dx, (5.23)


and

∂2P

∂s2= −

∫ρsD

′(ρ)v∇V dx−∫D(ρ)vs∇V dx

= −∫

div(D(ρ)v)D′(ρ)v∇V dx−∫D(ρ)∇

(12D′(ρ) |v|2

)∇V dx

=∫D(ρ)v∇(D′(ρ)v∇V ) dx−

∫D(ρ)D′(ρ)Jacob(v)v∇V dx

− 12

∫D(ρ)∇ρD′′(ρ)|v|2∇V dx

=∫D(ρ)|v|2∇ρD′′(ρ)∇V dx+

∫D(ρ)vD′(ρ)Jacob(v)∇V dx

+∫D(ρ)|v|2D′(ρ)Hess(V ) dx−

∫D(ρ)D′(ρ)Jacob(v)v∇V dx

− 12

∫D(ρ)∇ρD′′(ρ)|v|2∇V dx

=12

∫D(ρ)|v|2∇ρD′′(ρ)∇V dx+

∫D(ρ)|v|2D′(ρ)Hess(V ) dx.

The second term is nonnegative, if D′(ρ) ≥ 0 and Hess(V ) ≥ 0. However,nothing can be said about the sign of the first term, as the sign of ∇ρ isunknown and cannot be controlled by any other quantity.

5.2.3 System Case without Potential

In this section, we attempt to examine the displacement convexity of theentropy functional in the case when D(ρ) is matrix valued. We will call thisthe system case. The entropy is still defined via

E =∫u(ρ) dx, (5.24)

and we start again by calculating the first derivative.

dE

ds=

∫ρs · ∇ρ(u(ρ)) dx =

∫div(D(ρ)v) · ∇ρu(ρ) dx

= −∫ ∑

i,j,k

∂2u(ρ)∂ρi∂ρk

Dijvj · ∇ρk


Again, we define a function f , this time via

∂fi∂ρk

=∑i

∂2u(ρ)∂ρi∂ρk

Dij . (5.25)

Using this definition the above expression can be written as

dE

ds=∫ ∑

j

∇(fj) · vj . (5.26)

Next, we calculate the second derivative

d2E

ds2= −

∫ ∑j

∂ρj∂t

div(vj) dx+∫ ∑

j

∇(Fj) ·∂vj∂t

Unfortunately, in this general setting, we were not able to obtain any generalconditions ensuring that the entropy is displacement convex. To betterunderstand the properties of these terms we consider the following example.

5.2.4 Chen Model for Energy Transport

The Chen-Model for energy transport is given by the energy

u(ρ1, ρ2) =52ρ1 log(ρ1))−

32ρ1 log(ρ2), (5.27)

and the diffusion matrix

D(ρ1, ρ2) :=

(ρ1 ρ2

ρ253ρ22ρ1

). (5.28)

With this model, eq. 5.25 reduces to

∂fik∂ρk

= δjk, (5.29)

where δjk denotes the Kronecker delta. This furthermore simplyfies thesecond derivative to

d2E

ds2= −

∫ (∂ρ1

∂tdiv(v1) +

∂ρ2

∂tdiv(v2)

)dx− 1

2

∫ ∑i

∇ρi · ∇

∑j,k

∂Djk

∂ρivj · vk

dx.


Using the definition of the geodesic in the n-dimensional setting, eq. (3.48),we get

d2E

ds2= −

∫ (∂ρ1

∂tdiv(v1) +

∂ρ2

∂tdiv(v2)

)dx

−∫ (

12∇ρ1 · ∇

(v21 −

53g22

ρ21

v22

)− 1

2∇ρ2 · ∇

(2v1 · v2 +

103ρ2

ρ1v22

))dx

= −∫ (

∂ρ1

∂tdiv(v1) +

∂ρ2

∂tdiv(v2)

)dx

−∫

(−∇ρ1 · v1 −∇ρ2 · v2) div(v1)

+(

53ρ22

ρ21

∇ρ1 · v2 −103ρ2

ρ2∇ρ2 · v2 −∇ρ2 · v1

)div(v2)

+53∇(ρ22

ρ21

)· ∇ρ1v

22 −

53∇(ρ2

ρ1

)· ∇ρ2v

22 dx.

Now we use the second PDE to the geodesics, eq. (3.49) to explicitly calcu-late ∂ρ1

∂s and ∂ρ2∂s .(

ρ1

ρ2

)s

= div(D(ρ)v) = div

((ρ1 ρ2

ρ253ρ22ρ1

)(v11 v12

v21 v22

))

= div

((ρ1v11 + ρ2v21 ρ1v12 + ρ2v22

ρ2v11 + 53ρ22ρ1v21 ρ2v12 + 5

3ρ22ρ1v22

))

=

((ρ1v11)x1 + (ρ2v21)x1 + (ρ1v12)x2 + (ρ2v22)x2

(ρ2v11)x1 + (53ρ22ρ1v21)x1 + (ρ2v12)x2 + (5

3ρ22ρ1v22)x2

)

Inserting this into the equation for the second derivative and using the prod-uct rule we finally obtain

d2E

ds2=

∫ (ρ1div2(v1) + 2ρ2div(v1)div(v2) +

53ρ22

ρ1div2(v2)

)dx

+∫ (

56∇(ρ22

ρ21

)· ∇ρ1v

22 −

53∇(ρ2

ρ1

)· ∇ρ2v

22

)dx

=∫ √ρ1div(v1) +

√ρ22

ρ1div(v2)

2

+23ρ22

ρ1div2(v2) dx

−53

∫ (ρ2

ρ3/21

|∇ρ1|+1

ρ1/21

|∇ρ2|

)2

v22 dx.

5.3. Purely One-Dimensional Calculations 53

Unfortunately, in this result, the last term is negative and therefore we don’tknow the sign of the difference of the two terms. However, as the Chenmodel without potential reduces to the heat equation, it is very likely thatthe entropy is displacement convex. Thus, this results should be investigatedfurther in the future.

5.3 Purely One-Dimensional Calculations

In the above calculations it turned out that the entropy dissipation cal-culation gave different results that the displacement convexity calculation.Interestingly, this is not the case for a purely one-dimensional calculation.

5.3.1 Entropy Dissipation in One Dimension

Again, the entropy is given by

E(ρ) =∫u(ρ) + ρV (x),

where ρ, u and V are now all functions from R to R.For the first derivative we obtain

dE

dt=∫ρt(u′(ρ) + V (x)) dx = −

∫D(ρ)ξ2 with ξ := (u′)x + Vx

The second derivative is

−d2E

dt2=

∫ρtD

′(ρ)ξ2 dx+ 2∫D(ρ)ξξt dx

=∫

(D(ρ)ξ)xD′(ρ)ξ2 dx− 2∫

((D(ρ)ξ)x)2 dx

= −∫D(ρ)ξ(D′(ρ)ξ2)x dx− 2

∫((D(ρ)ξ)x)

2 dx

= −2∫D(ρ)D′(ρ)ξ2ξx dx−

∫D(ρ)D′′(ρ)ξ2ξρx dx− 2

∫((D(ρ)ξ)x)

2 dx

= −2∫D(ρ)D′(ρ)ξ2Vxx dx− 2

∫D(ρ)D′(ρ)ξ2(u′)xx dx−

∫D(ρ)D′′(ρ)ξ2ξρx dx

− 2∫

((D(ρ)ξ)x)2 dx

=: (1) + (2) + (3) + (4).


We will now have a closer look at each of these terms,

(2) = −2∫D(ρ)D′(ρ)ξ2(u′)xx dx = 2

∫(D(ρ)D′(ρ)ξ2)x(u′)x dx

= 2∫

(D′(ρ))2ξ2(ρx)2u′′(ρ) dx+ 2∫D(ρ)D′′(ρ)ξ2(ρx)2u′′(ρ) dx

+ 4∫D(ρ)D′(ρ)ξξxρxu′′(ρ) dx,

and

(4) = −2∫D2(ρ)(ξx)2u′′(ρ) dx− 2

∫(D′(ρ))2ξ2(ρx)2u′′(ρ) dx

− 4∫D(ρ)D′(ρ)ξxξρxu′′(ρ) dx.

Collecting terms this leads to

−d2E

dt2= −2

∫D(ρ)D′(ρ)ξ2Vxx dx+

((((((((((((((2∫

(D′(ρ))2ξ2(ρx)2u′′(ρ) dx

+ 2∫D(ρ)D′′(ρ)ξ2(ρx)2u′′(ρ) dx+

((((((((((((((

4∫D(ρ)D′(ρ)ξξxρxu′′(ρ) dx

−∫D(ρ)D′′(ρ)ξ2ξρx dx− 2

∫D2(ρ)(ξx)2u′′(ρ) dx−

((((((((((((((2∫

(D′(ρ))2ξ2(ρx)2u′′(ρ) dx

−((((((((((((((

4∫D(ρ)D′(ρ)ξxξρxu′′(ρ) dx.

The final result is then

−d2E

dt2= −2

∫D(ρ)D′(ρ)ξ2Vxx dx+ 2

∫D(ρ)D′′(ρ)ξ2(ρx)2u′′(ρ) dx


∫D2(ρ)(ξx)2u′′(ρ) dx.

Using the notation f ′ = D(ρ)u′′(ρ) this can be rewritten as

−d2E

dt2= −2

∫D(ρ)D′(ρ)ξ2Vxx dx+ 2

∫f ′(ρ)D′′(ρ)ξ2(ρx)2 dx


∫D(ρ)f ′(ρ)(ξx)2 dx

5.3. Purely One-Dimensional Calculations 55

5.3.2 Displacement Convexity in One Dimension

Here, the entropy is defined via

E(ρ) =∫u(ρ) + ρV (X) dx.

We calculate the derivatives of this entropy functional along geodesics whichare defined by the set of equations

ρs = (D(ρ)v)x

vs = −12(D′(ρ)v2).

For the first derivative we obtain

dE

ds=

∫ρs(u′(ρ) + V (x)) dx = −

∫D(ρ)v(u′)x dx−

∫D(ρ)vVx dx

= −∫f(ρ)xv dx−

∫D(ρ)vVx dx.

For the Second derivative the result is

d2E

ds2= −

∫(ρsf ′(ρ))xv dx−

∫f(ρ)xvx dx−

∫ρsD

′(ρ)vVx dx−∫D(ρ)vxVx dx

=∫

(D(ρ)v)xf ′(ρ)vx dx−12

∫f ′(ρ)ρx(D′(ρ)v2)x dx+

∫D(ρ)v(D′(ρ)vVx)x dx

− 12

∫D(ρ)(D′(ρ)v2)xVx dx.

Using product rule this leads to

d2E

ds2=

∫D′(ρ)f ′(ρ)ρxvvx dx+

∫D(ρ)f ′(ρ)(vx)2 −

12

∫f ′(ρ)D′′(ρ)(ρx)2(vx)2 dx

−∫D′(ρ)f ′(ρ)vvxρx dx+

∫D(ρ)D′′(ρ)v2ρxVx dx+

∫D(ρ)D′(ρ)vvxVx dx

+∫D(ρ)D′(ρ)v2Vxx dx−

12

∫D(ρ)D′′(ρ)v2ρxVx −

∫D(ρ)D′(ρvvxVx dx.

This leads to

d2E

ds2=

∫D(ρ)f ′(ρ)(vx)2 −

12

∫f ′(ρ)D′′(ρ)(ρx)2(vx)2 dx

+∫D(ρ)D′′(ρ)v2ρxVx dx+

∫D(ρ)D′(ρ)v2Vxx dx−

12

∫D(ρ)D′′(ρ)v2ρxVx dx


and thus

d2E

ds2=

∫D(ρ)f ′(ρ)(vx)2 −

12

∫f ′(ρ)D′′(ρ)(ρx)2(vx)2 dx

+∫D(ρ)D′(ρ)v2Vxx dx+

12

∫D(ρ)D′′(ρ)v2ρxVx dx.

5.4 Results

If we finally compare the results of the two calculations in one dimension,namely for the entropy dissipation

d2E

dt2= 2

∫D(ρ)D′(ρ)ξ2Vxx dx+

∫D(ρ)D′′(ρ)ξ2ρxVx dx

+ 2∫D(ρ)f ′(ρ)(ξx)2 dx−

∫D′′(ρ)f ′(ρ)ξ2(ρx)2 dx.

and for the displacement convexity

d2E

ds2=

∫D(ρ)D′(ρ)v2Vxx dx+

12

∫D(ρ)D′′(ρ)v2ρxVx dx

+∫D(ρ)f ′(ρ)(vx)2 dx−

12

∫D′′(ρ)f ′(ρ)(vx)2(ρx)2 dx,

we see that, despite a factor 1/2, both calculation agree.The differences that occur in displacement and entropy convexity calcula-tions in more than one dimension might be the consequence of the fact thatevery two points on the Riemannian manifold can always be connected via ageodesic while this is certainly not true for a solution of the PDE. Therefore,every curve defined by a solution could probably be approximated piecewiseusing geodesics. This would be done in such a way that the starting pointis chosen on the solution and the end point of the geodesic is determinedto that the slope of the geodesic and the slope of the solution agree for ashort time interval. To explore this argument more carefully would be aninteresting to understand the relation between the two types of convexityused in this thesis.

Chapter 6

Summary

This thesis addresses the asymptotic (or long-time) behaviour of solutionsto non-linear Fokker-Planck type PDEs. This is done using two differentmethods. Both techniques exploit convexity of an entropy functional whichcan naturally be defined for the given type of equation.The first idea is to calculate the derivatives of the entropy functional alongsolutions of the equation. It is observed that the first derivative is the nega-tive of some entropy production functional (which means that the entropy isa Lyaponov functional with respect to solutions of the PDE). This alreadyshows that the entropy is non-increasing along solutions. Looking at thesecond derivative it can be shown that the functional is even uniformly con-vex which ensures the existence of a minimum of the entropy and thus theexistence of a stationary state. However, this is only archived under certainassumptions on the non-linearity in the PDE which are further studied usingsome common examples.The second method is based on the observation that the PDE can be in-terpreted as a gradient flow on a certain (at least formally) Riemannianmanifold, with respect to the entropy functional. Therefore it is natural toexamine the convexity properties of the entropy along geodesics in this man-ifold. In the context of optimal transport, convexity along geodesics is calleddisplacement convexity. To clearify the connection to optimal transport ashort introduction is given in chapter 2. Again, it is shown that the entropydefines a Lyaponov functional and that, again under certain assumptions, itis uniformly displacement convex.However, it is observed that the conditions for both methods do not coin-

57

58 Chapter 6. Summary

cide, except for the one-dimensional case. The reason for this is not yetcompletely understood even though the fact that between any two pointson the manifold a geodesic connecting them can be found. This is certainlynot true for solutions of the PDE.Finally, an attempt has been made to examine the PDE for a matrix-valuenonlinearity. However, in this general setting, no statements could be made.

Appendix A

Some facts from Measure

Theory

We state some basic definitions and theorems concerning measure theory.

Definition (Metric space) A metric space is a set X together with a dis-tance function d : X ×X → [0,∞), having the following proporties:

d(x, y) = 0 ⇔ x = y (A.1)

d(x, y) = d(y, x) (A.2)

d(x, y) ≤ d(x, z) + d(z, y) ∀z ∈ X. (A.3)

(A.4)

Definition (Separable Metric space) A Metric Space (X, d) is called seper-able if it contains a countable dense subset.

Definition (Summable Functions) Let f be measurable and nonnegative.Then, f is called summable (or integrable) function if∫

fdµ <∞.

Definition (Lp spaces) Let Ω be a measureable space with a (positive)measure µ and let 1 ≤ p <∞. Then the Space Lp(Ω, dµ) is defined by

Lp(Ω, dµ) := f : f : Ω → (C), f is µ-measurable and |f |p is µ-summable.(A.5)

59

60 Chapter A. Some facts from Measure Theory

Furthermore, we define

L∞(Ω, dµ) := f : f : Ω → (C), f is µ-measurable and there exists a finite

constant K such that |f(x)| ≤ K for µ− a.e. x ∈ RN.

Theorem A.0.1 (Rademacher’s Theorem) If Ω ⊂ RN is open and f : Ω →Rm is Lipschitz, then f is differentiable almost everywhere.

Definition (Absolute Continous Measures) Let µ and ν be two measureson (X,F). Then ν is absolutely continous with respect to µ (write ν << µ)if

µ(S) = 0 ⇒ ν(S) = 0

for all S ⊂ F .

Theorem A.0.2 (Radon-Nikodym Theorem) Let µ and ν be two finitemeasures on (X,F). If ν is abolutely continous with respect to µ, then∫

XF (x)dν(x) =

∫XF (x)h(x)dµ(x) (A.6)

holds for some nonnegative h ∈ L1(X,µ) and every measurable F .

A.1 Convex Analysis

Definition (Lower semi-continuity) A function on a topological space X iscalled lower semi-continous, if it satisfies

F (x) ≤ lim infy→x

F (y). (A.7)

Theorem A.1.1 (Jenson’s inequality) Assume f : R → R is convex, andU ⊂ RN open, bounded. Let u : U → R be summable. Then

f

(1|U |

∫Uudx

)≤ 1|U |

∫Uf(u)dx. (A.8)

Appendix B

Some facts from Differential

Geometry

Here, we will recall some facts from differential geometry, mainly following[Bar06] and [Burns05].First, we introduce the concept of a topological Mannifold.

Definition (Topology)Let M be a set. A collection O of subsets of M iscalled topology if,

i) ∅, M ∈ O

ii) If Ui ∈ O, i ∈ I, then∪i∈IUi ∈ O

iii) If U1, U2 ∈ O, then U1 ∩ U2 ∈ O.

A Pair (M,O) is called a topological space. A set U ⊂M is called open inM, if U ∈ O. A subset is called closed, if M −A ∈ O.We now consider two topological spaces M and N and look at the mapf : M → N . We call f continous, if

f−1(V ) ∈ OM for all V ∈ ON . (B.1)

If f is one to one and f−1 is continous as well , we call f a homoomorphism.Furthermore, we call two topological spaces homeomorph, if there exists ahomeomorphism between them.No, we can define the central concept of a Mannifold.

61

62 Chapter B. Some facts from Differential Geometry

Definition (Topological Mannifold)Let (M,O) be a topological space. Mis called n-dimensional topological Mannifold if

i) M is hausdorff, i.e. for all p, q ∈ M with p 6= qthere exists open setsU, V ⊂M with p ∈ U , q ∈ V and U ∩ V = ∅.

ii) The topology on M has a countable basis, i.e. a countable subsetB ∈ O, such that for every U ∈ O there exist Bi ∈ B, i ∈ I with

U = ∪i∈IBi.

iii) M is locally homeomorph to RN , i.e. for all p ∈ M there exist anopen subset U ⊂ M with p ∈ U , an open subset V ⊂ RN and ahomeomorphism x : U → V .

Definition (Tangential Vector) Let M be a differentiable Mannifold andp ∈M . A tangential vector in the point p is definied as the aquivalence classof all differentiable curves c : (−ε, ε) →M such that ε ≥ 0 and c(0) = p. Theaquivalence relation for two curves c1 : (−ε1, ε1) →M and c : (−ε2, ε2) →M

is defined byd

dt(x c1) |t=0 =

d

dt(x c2) |t=0, (B.2)

for a given map x : U → V .

The next step is to introduce differentiation on Manifolds. The simplestidea would be, given a point p ∈ M to say that a function f : M → R isdifferentiable in p, if the composition f x−1 is differentiable, i.e. to use thewell-known differentiation on R. However, it is a priori not clear that thisdefinition does not depend on the choise of the map x.

Definition Let M be a n-dimensional topological Mannifold. Two mapsx : U → V and y : U → V a called C∞-compatible, if

y x−1 : x(U ∩ U) → y(U ∩ U)

is a C∞ diffeomorphism. Next, we call the set of maps xα : Uα → Vα, α ∈ A,an atlas, if

∪α∈AUα = M

. Finally, an atlas is called C∞-atlas if every two maps in it are C∞-compatible.

B.1. The Tangent Space 63

This directly leads to the definition of a differential structure.

Definition A C∞-atlas Amax is called maximal (or differential structure),if every map that is C∞-compatible with Amax is already in it.Furthermore, a pair (M,Amax), where M is a n-dimensional Mannifold iscalled n-dimensional differentiable Mannifold.

B.1 The Tangent Space

We know want to define the derivative of a differentiable map between twoMannifolds. This will lead to the concept of the tangent space. Furthermore,we call the set TpM := c(0)|c : (−ε, ε) → M differentiable with c(0) = p

the tangent space of M at the point p.

B.2 Riemannian Geometry

Definition (Metric Tensor) A Metric tensor is a bilinear form definied onthe tangent space TpM

gp(q, s) : TpM × TpM → R

which smoothly varies with the base point p.

Definition A Riemannian metric an a differentiable manifold M is given bya scalar product on each tangent space TpM which depends smoothly on thebase point p. A Riemannian manifold is a differential manifold, equippedwith a Riemannian metric.

64 Chapter B. Some facts from Differential Geometry

Bibliography

[Ambrosio00] L. Ambrosio. Lecture notes on optimal transport problems,2000. Lecture Notes in Mathematics “Mathematical aspectsof evolving interfaces” (CIME Series, Madeira (PT), 2000)1812, P. Colli and J.F. Rodrigues Eds., 1–52, 2003.

[Ambrosio05] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savare. GradientFlows in Metric Spaces and in the Space of Probability Mea-sures. Lectures in Mathematics ETH Zuerich. Birkhaeuser,Basel, Boston, Berlin, 2005. ISBN 3-7643-2428-7.

[Bar06] C. Bar. Differentialgeometrie, lecture notes. 2006.

[Benamou00] Jean-David Benamou and Yann Brenier. A computationalfluid mechanics solution to the Monge–Kantorovich masstransfer problem. 84(3):375–393, October 2000. ISSN 0029-599X (print), 0945-3245 (electronic).

[Burns05] Keith Burns and Marian Gidea. Differential Geometry andTopology: With a View to Dynamical Systems (Studies in Ad-vanced Mathematics). Chapman and Hall, 2005.

[Carillo01] Jose A. Carillo, Ansgar Jungel, Peter A. Markovich, GiuseppeToscani, and Andreas Unterreiter. Entropy dissipation meth-ods for degenerate parabolic problems and generalized sobolevinequalities. Monatsh. Math., 133:1–82, 2001.

[Carillo06] Jose A. Carillo, Robert J. McCann, and Cedric Villani. Con-tractions in the 2-wasserstein length space and thermalizationof granular media. Archive for Rational Mechanics and Anal-ysis, 179:217–263, 2006.

65

66 BIBLIOGRAPHY

[Csisz’ar67] I. Csisz’ar. Information-type measures of difference of prob-ability distributions and indirect observations. Studia Sci.Math. Hungar., 2:299–318, 1967.

[Evans01] Lawrence C. Evans. Partial differential equations and mon-gekantorovich mass transfer, 2001. Lecture Notes in Mathe-matics.

[Evans04] Lawrence C. Evans. A survey of entropy methods for partialdifferential equations. Bull. Amer. Math. Soc., 41:409–438,2004.

[Gronwall19] T. H. Gronwall. Note on the derivatives with respect to aparameter of the solutions of a system of differential equa-tions. The Annals of Mathematics, 20(4):292–296, 1919. ISSN0003486X.

[Kullback59] S. Kullback. Information theory and statistics. John Wileyand Sons., New York, 1959.

[McCann95] Robert McCann and Wilfrid Gangbo. Optimal maps inmonge’s mass transport problem. C.R. Acad. Sci. Paris. Ser.I. Math., 325:1653–1658, 1995.

[McCann97] Robert McCann. A convexity principle for interacting gases.Adv. Math., 128:153–179, 1997.

[Otto01] Felix Otto. The geometry of dissipative evolution equations:the porous medium equation. Comm. Partial DifferentialEquations, 26(1-2):101–174, 2001. ISSN 0360-5302.

[Otto05] Felix Otto and Michael Westdickenberg. Eulerian calculus forthe contraction in the Wasserstein distance. SIAM J. Math.Anal., 37(4):1227–1255 (electronic), 2005. ISSN 0036-1410.

[Pattle59] R. E. Pattle. Diffusion from an instantaneous point sourcewith a concentration-dependent coefficient. Quart. J. Mech.Appl. Math., 12:407–409, 1959.

[Risken89] Hannes Risken. The FokkerPlanck Equation: Methods of So-lutions and Applications, second ed. Springer Verlag, Berlin,1989.

BIBLIOGRAPHY 67

[Villani03] Cedric Villani. Topics in Optimal Transportation (GraduateStudies in Mathematics, Vol. 58). American MathematicalSociety, March 2003.

68 BIBLIOGRAPHY

Eigenstandigkeitserklarung

Ich versichere, diese Arbeit selbstandig verfasst und keine anderen als dieangegebenen Hilfsmittel und Quellen benutzt zu haben.

Munster, 4. Oktober 2008

Jan-Frederik Pietschmann

long-time behaviour of nonlinear fokker-planck equations · 2018-07-03 · formulae, we consider...

Documents