The Landau equation in a domain

TL;DR

This paper proves the global existence and decay to equilibrium of solutions to the Landau equation with Maxwell boundary conditions in a bounded domain, including Coulomb interactions, extending previous results to new boundary conditions.

Contribution

It establishes the first existence and decay results for the Landau equation with Maxwell reflection boundary conditions in bounded domains, covering the full interaction potential range.

Findings

Proved global existence of solutions in a bounded domain with Maxwell boundary conditions.

Established asymptotic decay to equilibrium for solutions close to Maxwellian.

Demonstrated hypocoercivity and ultracontractivity of the associated linear operator.

Abstract

This work deals with the Landau equation in a bounded domain with the Maxwell reflection condition on the boundary for any (possibly smoothly position dependent) accommodation coefficient and for the full range of interaction potentials, including the Coulomb case. We establish the global existence and a constructive asymptotic decay of solutions in a close-to-equilibrium regime. This is the first existence result for a Maxwell reflection condition on the boundary and that generalizes the similar results established for the Landau equation for other geometries in \cite{GuoLandau1,GS1,GS2,MR3625186,MR4076068}. We also answer to Villani's program \cite{MR2116276,MR2407976} about constructive accurate rate of convergence to the equilibrium {(quantitative H-Theorem)} for solutions to collisional kinetic equations satisfying a priori uniform bounds. The proofs rely on the study of a suitably linear problem for which we prove that the associated operator is hypocoercive, the associated semigroup is ultracontractive, and finally that it is asymptotically stable in many weighted $L^\infty$ spaces.