Well-posedness of the spatially homogeneous Landau equation for soft potentials

TL;DR

This paper establishes well-posedness for the spatially homogeneous Landau equation with soft potentials, using probabilistic methods and Wasserstein distance analysis, addressing singularities at small velocities.

Contribution

It provides the first uniqueness results for soft potentials and extends well-posedness to a broad class of initial conditions with finite mass, energy, and entropy.

Findings

First uniqueness result for soft potentials.

Well-posedness under finite mass, energy, and entropy.

Local well-posedness for very soft potentials.

Abstract

We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderatley soft potentials, we assume additionnally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka.