Half-space decay for linear kinetic equations

TL;DR

This paper establishes decay rates for solutions to linear kinetic equations in a half-space with absorbing boundaries, showing they decay faster than in the whole space and aligning with heat equation decay.

Contribution

It provides the first precise decay estimates for a broad class of linear kinetic equations in a half-space with absorbing boundary conditions.

Findings

Solutions decay as t^{-1/2 - d/4} in weighted L^2 space.

Solutions decay as t^{-1 - d/2} in weighted L^∞ space.

Decay rates match those of the heat equation with Dirichlet conditions.

Abstract

We prove that solutions to linear kinetic equations in a half-space with absorbing boundary conditions decay for large times like $t^{-\frac{1}{2}-\frac{d}{4}}$ in a weighted $\sfL^{2}$ space and like $t^{-1-\frac{d}{2}}$ in a weighted $\sfL^{\infty}$ space, i.e. faster than in the whole space and in agreement with the decay of solutions to the heat equation in the half-space with Dirichlet conditions. The class of linear kinetic equations considered includes the linear relaxation equation, the kinetic Fokker-Planck equation and the Kolmogorov equation associated with the time-integrated spherical Brownian motion.