On the gradient flow structure of the isotropic Landau equation

TL;DR

This paper demonstrates that the isotropic Landau equation with Coulomb potential can be viewed as a gradient flow of entropy in a probability space, providing new insights into its geometric structure and convergence properties.

Contribution

It establishes the gradient flow structure of the isotropic Landau equation with Coulomb potential and characterizes its geodesics and convergence rate.

Findings

Identification of the Landau equation as a gradient flow of entropy

Characterization of geodesic equations in the associated metric space

Estimation of convergence rate via Hessian operator analysis

Abstract

We prove that the isotropic Landau equation equipped with the Coulomb potential introduced by Krieger-Strain and Gualdani-Guillen can be identified with the gradient flow of the entropy in the probability space with respect to a Riemannian metric tensor with nonlocal mobility. We give characterizations of the corresponding geodesics equations and present a convergence rate result by estimating its Hessian operator.