Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

Armand Bernou (1) , Kleber Carrapatoso (2) , Stéphane Mischler (3) , Isabelle Tristani (4)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CMLS - Centre de Mathématiques Laurent Schwartz
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 DMA - Département de Mathématiques et Applications - ENS Paris

We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator, in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all type of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition.

Domains: Mathematics [math]; Analysis of PDEs [math.AP]

Armand Bernou, Kleber Carrapatoso, Stéphane Mischler, Isabelle Tristani. Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, In press, ⟨10.4171/AIHPC/44⟩. ⟨hal-03142785⟩