Uniqueness of higher integrable solution to the Landau equation with Coulomb interactions

TL;DR

This paper proves the uniqueness of weak solutions to the Landau equation with Coulomb interactions under certain boundedness conditions, utilizing a weighted Poincaré-Sobolev inequality.

Contribution

It establishes the uniqueness of solutions in a specific function space, extending previous results with a novel inequality approach.

Findings

Uniqueness holds for solutions bounded in L^(0,T,L^p(\u211d^3)) with p>3/2.

The proof employs a recently introduced weighted Poincare9-Sobolev inequality.

Results contribute to the mathematical understanding of Coulomb-interacting particle systems.

Abstract

We are concerned with the uniqueness of weak solution to the spatially homogeneous Landau equation with Coulomb interactions under the assumption that the solution is bounded in the space $L^\infty(0,T,L^p(\R^3))$ for some $p>3/2$. The proof uses a weighted Poincar\'e-Sobolev inequality recently introduced in \cite{GG18}.