Hypocoercivity and sub-exponential local equilibria

Abstract

Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the heat equation obtained in the diffusion limit. The method applies to Fokker-Planck and scattering collision operators. The main tools are a weighted Poincaré inequality (in the Fokker-Planck case) and norms with various weights. The advantage of weighted Poincaré inequalities compared to the more classical weak Poincaré inequalities is that the description of the convergence rates to the local equilibrium does not require extra regularity assumptions to cover the transition from super-exponential and exponential local equilibria to sub-exponential local equilibria.

Nonlinear stability of chemotactic clustering with discontinuous advection

“…We are not aware of the occurrence of (3.1) in the literature. There are some examples of weighted Poincaré inequalities with non standard averaging, see for instance [4,Appendix A.2]. We can also report the extremal case λ = +∞ which coincides with the following Hardy inequality:…”

Hypocoercivity and global hypoellipticity for the kinetic Fokker-Planck equation in $H^k$ spaces

“…Moreover, hypocoercivity can be measured in many different ways, such as entropy, ϕentropies, total variation, Wasserstein distance, L 2 norms, H k norms, or more general weighted Sobolev norms, see for instance [5], [9], [3], [41], [16], [21], [7], [24](see also related work [22] and [23]), [25] and the references therein. We also call attentions to [39], [18], [10], [11], [12], [15], [14] and [30] for some recent results concerning hypocoercivity with various kinds of potentials. In particular, we highlight [8] and [37] for some results concerning singular potentials.…”