Convergence toward the steady state of a collisionless gas with Cercignani–Lampis boundary condition

Abstract

We study the asymptotic behavior of the kinetic free-transport equation enclosed in a regular domain, on which no symmetry assumption is made, with Cercignani-Lampis boundary condition. We give the first proof of existence of a steady state in the case where the temperature at the wall varies, and derive the optimal rate of convergence towards it, in the L 1 norm. The strategy is an application of a deterministic version of Harris' subgeometric theorem, in the spirit of [10] and [4]. We also investigate rigorously the velocity flow of a model mixing pure diffuse and Cercignani-Lampis boundary conditions with variable temperature, for which we derive an explicit form for the steady state, providing new insights on the role of the Cercignani-Lampis boundary condition in this problem. Acknowledgements:The author acknowledges financial support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement number 864066).

On quantitative hypocoercivity estimates based on Harris-type theorems

“…In [8,9], the author gave the first quantitative subgeometric convergence results on this equation, with boundary conditions, in dimensions d ∈ {2, 3} by using Harris's theorem. We refer also e.g.…”

“…In this section we will present the main results of [8,9]. First, we define a map τ which gives the time of the first collision with the boundary for a particle in the phase space z…”

“…The main results of [8,9] are given by Theorem 3.4. Suppose that t → f t is the solution to the free-transport equation…”

“…We give the sketch of the proofs separately for two regimes. For the details we refer to [8,9] for R1 and R2 respectively.…”

“…The proof is similar to the case of R1 except two key points. We remark here only the differences and we refer to [9] for the details. Similar to (3.37) we would like to obtain…”