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

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.