Special macroscopic modes and hypocoercivity

TL;DR

This paper classifies special macroscopic modes in kinetic equations with multiple conservation laws and proves exponential convergence of solutions to these modes, providing the first hypocoercivity results with constructive estimates based on potential geometry.

Contribution

It introduces a comprehensive classification of macroscopic modes and establishes quantitative hypocoercivity results for equations with multiple conservation laws.

Findings

Classification of all special macroscopic modes including stationary and periodic solutions

Proof of exponential convergence of solutions to these modes

Constructive estimates depending on the potential's geometry

Abstract

We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator admitting several local conservation laws (local density, momentum and energy). We classify all special macroscopic modes (stationary solutions and time-periodic solutions). We also prove the convergence of all solutions of the evolution equation to such non-trivial modes, with a quantitative exponential rate. This is the first hypocoercivity result with multiple special macroscopic modes with constructive estimates depending on the geometry of the potential.