Boltzmann to Landau from the Gradient Flow Perspective

TL;DR

This paper explores the connection between the Boltzmann and Landau equations through the lens of gradient flows, using a $ ext{Gamma}$-convergence approach to analyze their entropy-dissipation behavior during the grazing collision limit.

Contribution

It introduces a novel gradient flow perspective on the Boltzmann to Landau limit, employing $ ext{Gamma}$-convergence techniques for entropy-dissipation convergence.

Findings

Established lower semi-continuous convergence of entropy-dissipation

Reinterpreted the grazing collision limit via gradient flows

Extended the $ ext{Gamma}$-convergence framework to kinetic equations

Abstract

We revisit the grazing collision limit connecting the Boltzmann equation to the Landau(-Fokker-Planck) equation from their recent reinterpretations as gradient flows. Our results are in the same spirit as the $\Gamma$-convergence of gradient flows technique introduced by Sandier and Serfaty. In this setting, the grazing collision limit reduces to showing the lower semi-continuous convergence of the Boltzmann entropy-dissipation to the Landau entropy-dissipation.