Uniqueness of bounded solutions to the fuzzy Landau and multiespecies Landau equations

TL;DR

This paper establishes the uniqueness of weak solutions for the fuzzy Landau and multiespecies Landau equations using stability estimates, unifying several classical models within a broader framework.

Contribution

It introduces a novel approach based on stochastic coupling and symmetrization techniques to prove uniqueness for a class of nonlinear equations with singular coefficients.

Findings

Proves uniqueness of weak solutions under integrability assumptions.

Provides explicit stability estimates in the 2-Wasserstein distance.

Unifies several classical models like Euler, Vlasov-Poisson, and Patlak-Keller-Segel.

Abstract

We prove uniqueness of weak solutions to the fuzzy Landau equation and the multiespecies Landau system under suitable integrability assumptions. The results are based on explicit stability estimates in the 2-Wasserstein distance for a broader class of nonlinear equations with singular coefficients. Interestingly, this class includes the 2D incompressible Euler equations, the Vlasov-Poisson system, and the Patlak-Keller-Segel model, thereby recovering known uniqueness results within a unified framework. Our approach builds on the stochastic coupling method introduced by Fournier and Guerin for the homogeneous Landau equation, which we recast in a more analytic form. In addition, we present an alternative argument based on the symmetrization technique of Guillen and Silvestre, yielding comparable stability estimates.