I have uploaded a new paper entitled “Hypocoercivity for linear kinetic equations conserving mass”, written in collaboration with Jean Dolbeault from Paris-Dauphine and Christian Schmeiser from Wien University:
This paper deals with hypocoercivity, i.e. obtaining (constructive) rates of exponential decay for linear equations writing as a skew-symmetric part + a partially coercive part. Coercivity is degenerated but can be recovered taking advantage of the commutators with the skew-symmetric. Hence the name “hypocoercivity” by analogy with the hypoelliptic theory of Hormander. This is applied to Fokker-Planck and relaxation models in kinetic theory, and we provide the first theory for these models (without any regularity assumption on the initial data), the first unified theory for diffusive and integral collision models, and the first results for kinetic version of the fast diffusion equation (whose Gibbs state does not separate variables).
Here is the abstract:
We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models like the linear Boltzmann equation or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.