Uniform convergence to equilibrium for granular media

Introduction

We consider the problem of convergence to equilibrium for the nonlinear equation

Steady states may exist as a result of a balance between these three effects, and we are concerned with deriving rates of convergence of solutions towards them. Following , this issue has raised much attention in the last years and has been tackled by a particle approximation and logarithmic Sobolev inequalities in , by an entropy dissipation method in and by contraction properties in Wasserstein distance in (see also for related works in dimension one). The entropy method is based on studying the time derivatives of a Lyapunov function $F$ of the equation (called entropy or energy), on the interpretation due to F. Otto of (1) as a gradient flow of $F$ (see ) and on the notion of convexity for $F$ due to R. J. McCann (see ).

When $V$ and $W$ are uniformly convex, solutions converge exponentially fast to equilibrium, but the case of interest of is $V=0$ and $W(x)=|x|^{3}$, whose convexity degenerates at . For this case, only a polynomial rate, or exponential but depending on the initial data, was obtained in . In the present paper we prove a uniform exponential convergence in Wasserstein distance of all solutions to the steady state. The method, introduced in the linear case in , is based on comparing the Wasserstein distance with its dissipation along the evolution.

In Section 1 we derive the dissipation of the Wasserstein distance between solutions and easily deduce the classical contraction results. Section 2 is devoted to cases when the convergence is driven by the interaction potential $W$, with or without exterior potential $V$: in particular we prove the first result of uniform exponential convergence to equilibrium for degenerately convex interaction potentials and no exterior potential. In Section 3 we give conditions to get an exponential convergence to equilibrium with both potentials being non convex.

Dissipation of the Wasserstein distance

and $\nabla\varphi^{*}\#\nu=\mu$ for the Legendre transform $\varphi^{*}$ of $\varphi$ if also $\nu$ is absolutely continuous with respect to the Lebesgue measure. We refer to for instance for these notions.

for almost every $t>0$ and all probability measure $\sigma$ in the domain of $F$. For all $t>0$ the solution $\mu_{t}$ has a density with respect to the Lebesgue measure. Moreover the curve $\mu$ satisfies the continuity equation

in the sense of distributions, where the velocity field $v_{t}$ satisfies

If $(\mu_{t})_{t}$ and $(\nu_{t})_{t}$ are two solutions to (1), then for a.e. $t>0$,

where, for $\nu=\nabla\varphi\#\mu$ (and $\Delta\varphi$ the trace of the Hessian of $\varphi$ in the Alexandrov sense),

For $t>0$ we can expect the solutions to have smooth densities, and to have equality in Proposition 1.1, but we shall be content with the inequality (see ).

Considering the dissipation of the distance between two solutions provides simple alternative proofs of contraction properties in Wasserstein distance derived in . For that purpose we first notice that given $\mu$ and $\nu$ absolutely continuous with respect to the Lebesgue measure, and $\nabla\varphi\#\mu=\nu$, then $\Delta\varphi+\Delta\varphi^{*}(\nabla\varphi)-2n\geqslant 0$ $\mu$ a.e. (see for example [14, Th. 1.5] and [4, Lem. 2.5]). This inequality says that the diffusion part of the equation always contracts two solutions, as it is classical for the pure heat equation. Then:

since $\beta\leq 0$. Hence, for two solutions $(\mu_{t})_{t}$ and $(\nu_{t})_{t}$ of (1) and almost all $t\geqslant 0$

Then by the Gronwall lemma we recover the contraction property of [8, Th. 5]:

Suppose that $W$ is convex and that there exist $p,C>0$ such that for all $\varepsilon>0$

We optimize in $\varepsilon$ and integrate to recover the polynomial contraction of [8, Th. 6]

The following two sections are devoted to the obtention of explicit exponential rates of convergence of solutions to (1) in non uniformly convex or even non convex cases, having in mind the degenerately convex potentials of and the double well potentials of .

Influence of the interaction potential

In this section we study the case when $W$ brings the convergence.

We first assume that $V=0$. Then the evolution preserves the center of mass, and a solution $\mu_{t}$ should converge to a stationary solution $\mu_{\infty}$ only if the initial datum $\mu_{0}$ and $\mu_{\infty}$ have same center of mass, since

should converge to : for instance it is bounded by $W_{2}(\mu_{t},\mu_{\infty})$. We could also assume that $V\neq 0$, but that the center of mass is fixed by the evolution, which is all we use. But to simplify the statements we assume $V=0$.

When $W$ is degenerately convex, with a pointwise degeneracy, for instance $W(x)=|x|^{2+\varepsilon}$ with $\varepsilon>0$, then the contraction property holds only with polynomial decay rate, see the last example in Section 1. Then in the authors proved an exponential convergence to equilibrium, but not with a uniform decay rate, but rather depending on the free energy $F$ of the initial datum. In this section we prove a uniform exponential convergence for such potentials.

Then there exists an explicit positive constant $C$, depending only on $K,R$ and $M$, such that

for all measures $\nu$ with same center of mass as $\mu$.

Hypothesis (8) on $U$ holds on any of the following two instances:

$U$ is $\mathcal{C}^{2}$, $\nabla^{2}U(x)\geqslant\alpha(x)$ with $\alpha(x)\leq 0$ and $U(x)+2R^{2}\displaystyle\inf_{|x-y|\leq R}\alpha(y)\geqslant-M$ for all $x$; for example, $U$ is $\mathcal{C}^{2}$ and bounded from below and $\nabla^{2}U(x)\geqslant\alpha$ for all $x$ and a constant $\alpha$.

For $ii.$ for instance, assume that the sup of $U$ on $[x,y]$ is achieved at $z=tx+(1-t)y$ with $t\in]0,1[$. Then $\nabla U(z)\cdot(y-x)=0$, so that

since, by assumption on $\varphi$, the difference is

if $|x|\geqslant 2R$ or $|y|\geqslant 2R$. In view of this result we let

1. First of all, by convexity of $W$ and (9),

We let $H=\nabla^{2}\varphi(x+r\,t\,\theta)$ and write $H-I=[H^{1/2}-H^{-1/2}]H^{1/2}$, so that

by the Cauchy-Schwarz inequality. On the one hand, letting $D=\Delta\varphi+\Delta\varphi^{*}(\nabla\varphi)-2n$,

by (8). Now, for fixed $t\in$, the change of variables $(x,y)\mapsto(v,u)=(x+t(y-x),y-x)$ has unit Jacobian, so this is equal to

3. Collecting the terms in 1. and 2. concludes the proof with $C=2(\frac{3}{K}+4c_{n}R^{2+n}e^{M})^{-1}.$ $\rhd$

2 In presence of an exterior potential

We saw in Section 1 how an exterior potential $V$ can induce the convergence of all solutions to a unique equilibrium, and not only to the unique equilibrium with same center of mass as the initial datum of the solution to be considered.

If W strictly convex (but at ), and uniformly at infinity, and if $V$ is strictly convex (but at ), then polynomial convergence holds to a unique equilibrium $\mu_{\infty}$ (see Section 1 and [7, Th. 2.3]), and even exponential convergence, but with a rate depending on the free energy $F$ of the initial datum (see [7, Th. 2.5]). Following Theorem 2.1 (for $V=0$), one may wonder whether this convergence is actually uniform in the initial datum, given by

for all solution $(\mu_{t})_{t}$. But, according to Section 1, this estimate is based on the inequality

for the measure $\mu=\mu_{\infty}$ and all measures $\nu$; and this inequality does not hold if $V$ is only assumed to be strictly convex. For instance:

$\lhd$ We prove that (10) does not hold for the translations $\nu=\varphi^{\prime}\#\mu$ where $\varphi^{\prime}(x)=x+M$ where $M\to+\infty$, that is, that there is no $C>0$ such that

for all $M>0$. For that, we let $R$ to be fixed later on, and bound the right-hand side in (11) by

First of all, since $|V^{\prime\prime}|\leq A$, then $|V^{\prime}(x)|\leq|V^{\prime}(0)|+A|x|$ so the first integral is finite (uniformly in $M$), and the second one is bounded by

Now, for fixed $\varepsilon>0$, we take $R$ such that this is bounded by $(M+1)\varepsilon$ for all $M$. Then we take $M_{0}$ such that $|V^{\prime\prime}(x)|\leq\varepsilon$ for $x\geqslant M_{0}$. For all $M\geqslant M_{0}+R$ the third integral is bounded by

Collecting all terms we conclude that the full right-hand side in (11) is $\leq 4\varepsilon$ for large $M$. $\rhd$

Hence we can not expect a uniform rate of convergence to equilibrium for degenerately convex potentials. Our method is however able to recover an exponential convergence with a rate depending on the initial datum, as in [7, Th. 2.5]:

By Proposition 2.6 below, applied with the constant $N$ and the measure $\mu_{\infty}$,

Then, by the proof of Proposition 2.2, there exists $C_{1}$, depending only on $V$ and $W$, such that

Hence there exists a new positive constant $C$, depending only on $V,W$ and $M$, such that

$\lhd$ Let $S\geqslant 3R$ to be fixed later on. Since $V$ is $\mathcal{C}^{2}$ and $\nabla^{2}V(x)$ is definite positive on the compact set $R\leq|x|\leq S$, there exists $K=K(S)>0$ such that $\nabla^{2}V(x)\geqslant K$ for all $R\leq|x|\leq S$. Then, following [4, Lem. 5.1],

if $|x|\leq S$, $|y|\leq S$ and if $|x|\geqslant 2R$ or $|y|\geqslant 2R$; indeed one only need to take into account the values of $\nabla^{2}V$ on the ball of radius $S$.

1. First of all, by convexity of $V$, the above remark and the Cauchy-Schwarz inequality,

By the Cauchy-Schwarz and Markov inequalities, the first term is bounded from above by

Then, following the proof of [4, Prop. 3.5], there exists a constant $C$, depending on $V$ and $\mu$ only on the ball of radius $3R$, such that

Then we let S=\max\Big{\{}3R,\sqrt{12}\Big{[}\displaystyle\int|x|^{2}d\mu+N\Big{]}\Big{\}} so that

if $\int|\nabla\varphi|^{2}d\mu\leq N$, concluding the proof with a $C$ depending on $V,\mu$ and $M$ through $K(S)$. $\rhd$

Non convex examples

In this section we deal with potentials $V$ and $W$ for which the convergence rate to equilibrium is driven by $V$ rather than by $W$. Our first result is more qualitative rather than quantitative.

Assume that $V$ and $W$ are $\mathcal{C}^{2}$ convex maps and that there exist $R\geq 0$ and $K>0$ such that for all $|x|\geq R$,

In the first section (second example) we saw that only polynomial decay in contraction is known in this context, and only when the convexity degenerates at some points, for instance for $V(x)=|x|^{4}$.

First of all, a stationary solution, solution of $\mu_{\infty}^{\varepsilon}=e^{-V^{\varepsilon}-\mu_{\infty}^{\varepsilon}*W}/Z^{\varepsilon}$, exists by Proposition A.1, $iii$. Then one can then easily build a cut-off function $\psi$ such that $V^{\varepsilon}\psi$ is $\mathcal{C}^{2}$, convex, satisfies $(V^{\varepsilon}\psi)^{\prime\prime}\geqslant K>0$ outside a centered ball, uniformly in $\varepsilon\in$, and is such that $\|(V^{\varepsilon}(1-\psi))^{\prime\prime}\|_{\infty}$ converges to 0 as $\varepsilon\to 0$. Then, by [4, Prop. 3.5], the measure $\mu_{\infty}^{\varepsilon}$ satisfies a $WJ_{V^{\varepsilon}\psi,0}$ inequality with a constant $C>0$ uniformly in $\varepsilon\in$ (here we use that $\int Wd\mu_{\infty}^{\varepsilon}$ and $Z^{\varepsilon}$ are bounded uniformly in $\varepsilon$). Now the perturbation proposition [4, Prop. 3.8] ensures that $\mu_{\infty}^{\varepsilon}$ satisfies a $WJ_{V^{\varepsilon},0}$ inequality, for sufficiently small $\varepsilon$, hence a $WJ_{V^{\varepsilon},W}$ inequality since $W$ is convex. Here we say that a measure $\mu$ satisfies a $WJ_{V,W}$ inequality is the inequality (10) holds for a positive constant $C$ and all $\nu$.

The smallness condition on $\varepsilon$ is necessary since, according to , there exist several stationary solutions for large $\varepsilon$.

The following theorem provides the first examples of exponential convergence to equilibrium for the granular media equation, with both potentials non convex.

there exist $K\geqslant 0$ and $\beta\leq 0$ such that $\sup|W|\leq K$ and $\nabla^{2}W\geq\beta$.

Assumption (12) on the measure $e^{-V}$ has been studied in under the name of $WJ(C)$ inequality; there practical criteria have been given for the inequality to hold. Observe that we can always assume that $C\geqslant\alpha$ since, if $\alpha\geqslant 0$, then $\mu$ satisfies a $WJ(\alpha)$ inequality.

Then we let $d\mu(x)=e^{-V(x)}\,dx$ and use the convexity assumptions on $V$ and $W$, the bound on $W$ and the sign conditions on $\beta$ and $C-\alpha$ to get, for all $\nu=\nabla\varphi\#\mu_{\infty}$,

Appendix A Existence of stationary solutions

The existence of a minimizer of $F$ has been proved by R. J. McCann for strictly convex or radially symmetric convex interaction potentials $W$ (and $V=0$). We adapt his classical compactness-lower semicontinuity argument to our diverse cases:

$V=0$, $W$ is convex and $W(x)\geqslant b|x|^{2}-b^{\prime}$ for $b,b^{\prime}>0$;

$V(x)\geqslant a|x|-a^{\prime}$ and $W(x)\geqslant b|x|^{2}-b^{\prime}$ for $a,a^{\prime},b,b^{\prime}>0$;

$V(x)\geqslant a|x|^{2}-a^{\prime}$ and $W(x)\geqslant b|x|^{2}-b^{\prime}$ for $b^{\prime},a,a^{\prime}>0,b>-a$;

It remains now to bound $\int|x|^{2}d\mu_{p}$ by $F(\mu_{p})$ in each case:

For $i.$, as in , let $\nabla\varphi_{p}$ transport $\mu_{p}$ onto $\mu_{p}(-.)$ and let ${\bar{\mu}_{p}}=\frac{I+\nabla\varphi_{p}}{2}\#\mu_{p}$ for $I$ the identity map. Now $W$ is convex, so $F$ is displacement convex, so that $F({\bar{\mu}_{p}})\leq(F(\mu_{p})+F(\mu_{p}(-.)))/2=F(\mu_{p})$ and $({\bar{\mu}_{p}})$ is also a minimizing sequence. Moreover $\int xd{\bar{\mu}_{p}}=0$ so

hence $\int|x|d\mu_{p}$ is bounded by the second inequality, and then $\int|x|^{2}d\mu_{p}$ by the first one.

For $iii.$ we similarly observe, and by discussing on the sign of $b$, that

Acknowledgements. This research was supported by the French ANR project EVOL.

References