On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

TL;DR

This paper investigates the rate of convergence to equilibrium in the L^1 norm for solutions of the spatially homogeneous Boltzmann equation with soft potentials, establishing conditions for convergence and bounds on the convergence rate.

Contribution

It provides new results on the convergence to equilibrium for soft potentials, including time-averaged convergence, rate estimates based on initial energy tails, and bounds under angular cutoff.

Findings

Time-averaged L^1 convergence for solutions with finite entropy and moments

Convergence rate can be arbitrarily slow for initial data with long energy tails

Algebraic bounds on convergence rate under angular cutoff with -1 ≤ γ < 0

Abstract

The paper concerns $L^1$- convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials $(-4\le \gm<0$), with and without angular cutoff. We prove the time-averaged $L^1$-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than $2+|\gm|$. For the usual $L^1$-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cutoff on the collision kernel with $-1\le \gm<0$, there are algebraic upper and lower bounds on the rate of $L^1$-convergence to equilibrium. Our methods of proof are based on entropy inequalities and moment estimates.