On measure solutions of the Boltzmann equation, Part II: Rate of convergence to equilibrium

TL;DR

This paper establishes the exponential rate of convergence to equilibrium for measure solutions of the spatially homogeneous Boltzmann equation with hard potentials, providing bounds and stability results.

Contribution

It proves the sharp exponential convergence rate for measure solutions with finite mass and energy, extending previous results to measure solutions and analyzing their stability.

Findings

Exponential convergence rate to equilibrium established

Lower bounds on convergence rate derived

Global-in-time strong stability of solutions demonstrated

Abstract

The paper considers the convergence to equilibrium for measure solutions of the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. We prove the exponential sharp rate of strong convergence to equilibrium for conservative measure solutions having finite mass and energy. The proof is based on the regularizing property of the iterated collision operators, exponential moment production estimates, and some previous results on the exponential rate of strong convergence to equilibrium for square integrable initial data. We also obtain a lower bound of the convergence rate and deduce that no eternal solutions exist apart from the trivial stationary solutions given by the Maxwellian equilibrium. We finally use these convergence rates in order to deduce global-in-time strong stability of measure solutions.