Long-Time Behavior of Kinetic Equations with Boundary Effects

This thesis studies the long-time behavior of several partial differential equations arising in kinetic theory. Those equations have an equilibrium towards which, roughly, some solution converges. We employ both probabilistic and deterministic methods to derive the rate of this convergence. This work is divided into three parts. In the first one, we study the free-transport equation enclosed in a bounded domain with Maxwell boundary condition, as already considered by Aoki and Golse (2011) and Kuo et al. (2013, 2014, 2015). We extend the (almost) optimal rate , where is the dimension of the problem, to the case of a general regular domain, without the symmetry assumption required in earlier works. We use two different methods, first a probabilistic coupling in Chapter 2, allowing us to generalize the boundary condition, and a deterministic version of some subgeometric Harris’ theorem in Chapter 4, with which we can consider the case where the temperature varies at the boundary. In Chapter 3 we provide some numerical evidences supporting our result that the polynomial rate of convergence observed in the case where the spatial domain is symmetric should extend to non-symmetric, regular domains. In the second part of this thesis, we focus on the subgeometric convergence towards the invariant distribution of Markov processes. We exhibit a new set of conditions, close in spirit to the ones of Douc, Fort and Guillin (2009) and Hairer (2016), leading to the subgeometric convergence of a strong Markov process. Our conditions are chosen in order to be equivalent, as in the exponential theory of Meyn and Tweedie (1993), whereas only one implication holds in the usual set of conditions. In the last part, we study collisional kinetic models, namely the linearized Boltzmann and linearized Landau equations, enclosed in a regular, bounded domain. We prove constructive hypocoercivity estimates for the generalized Maxwell boundary condition, which includes the case of the specular reflection boundary condition. With those estimates, one concludes to the exponential relaxation towards equilibrium for those models.

Preprint

PhD Thesis, Sorbonne Université