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Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. (English) Zbl 1073.35127
The authors study asymptotic behaviour of the equation \[ \frac{\partial \rho}{\partial t}=\nabla\cdot\left(\rho\nabla(U'(\rho)+V+W*\rho)\right), \] where the unknown \(\rho\) is a probability measure on \(\mathbb{R}^d\), \(d\geq 1\), \(U\) is the density of internal energy, \(V\) is a confinement potential and \(W\) is an interaction potential. The functional \[ F(\rho) = \int_{\mathbb{R}^d}U(\rho)\,dx +\int_{\mathbb{R}^d}V(x)\,d\rho(x) +\frac{1}{2}\int_{\mathbb{R}^d\times \mathbb{R}^d}W(x-y)\,d\rho(x)d\rho(y) \] is the entropy, or free energy, associated with the equation (1). In many cases the competition between the potential determines a unique minimizer \(\rho_\infty\) for \(F\). The authors find conditions on \(U,V,W\) ensuring that this is true and proceed to determine whether the solutions to (1) converge to \(\rho_\infty\), and how fast. The rate of convergence is measured in terms of the relative free energy \(F(\rho| \rho_\infty)=F(\rho)-F(\rho_\infty)\). The treatment of the problem is fairly comprehensive and, following the papers, can be summarized in the heuristic rules:
Rule 1: A uniformly convex confinement potential implies an exponential decay to equilibrium. Moreover, if the convexity of the confinement potential is strong enough, it can overcome a lack of convexity of the interaction potential. Rule 2: When the center of mass is fixed, then a uniformly convex interaction potential implies an exponential decay to equilibrium.
Rule 3: When the interaction potential in only degenerately convex and there is no diffusion, then the decay is in general only algebraic.
Rule 4: In presence of (linear or superlinear) diffusion, a degenerately convex interaction potential induces an exponential trend to equilibrium.
Rule 5: In presence of diffusion, the interaction potential is able to drive the system to equilibrium even if the center of mass is moving.
The proofs use the energy dissipation method which, due to the structure of (1), can be applied here via the theory of logarithmic Sobolev inequalities. The authors use two approaches for proving the necessary inequalities: one is based on Bakry and Emery method [D. Bakry and M. Emery, In: Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 177–206 (1985; Zbl 0561.60080)] and the other uses the so called HWI method, which involves interpolation inequalities relating the entropy functional, Wasserstein metric and Fisher information, introduced by Otto and Villani [F. Otto and C. Villani, J. Funct. Anal., 173, 361–400 (2000; Zbl 0985.58019)]. The authors point out that the HWI approach, when applicable, is preferable to the Bakry-Emery argument as it is more direct and does not require a priori knowledge that \(\rho\to\rho_\infty\). However, the latter does not use explicitly mass transportation and allows to avoid difficulties related to smoothness issues associated with it.

MSC:
35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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References:
[1] Arnold, A., Markowich, P., Toscani, G. and Unterreiter, A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. Partial Differential Equations 26 (2001), no. 1-2, 43-100. · Zbl 0982.35113
[2] Bakry, D. and Emery, M.: Diffusions hypercontractives. In Séminaire de probabilités, XIX, 1983/84, 177-206. Lect. Notes in Math. 1123, Springer, 1985. · Zbl 0561.60080
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