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

F. -U. Caja-Lopez

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.

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.

Paper Structure

This paper contains 11 sections, 17 theorems, 189 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the proof
  4. A general uniqueness result
  5. Main computation
  6. Estimates involving the coefficients
  7. Existence of the curve of transport plans
  8. The case $0< \alpha < 2$
  9. Symmetrization method
  10. Fuzzy Landau equation
  11. Multiespecies Landau equation

Key Result

Theorem 1.1

Consider $f,g$ weak solutions of eq:fuzzy_Landau for $-3\leq \gamma \leq -2$, and let $\sqrt{\kappa}$ be bounded and Lipschitz. Assume that $f,g\in L^1_t L^p_v L^1_x$ for $p=\frac{3}{3+\gamma}\in [3,\infty]$. Then, we have where $\omega:[0,\infty)\longrightarrow[0,\infty)$ is continuous, increasing, and satisfies $\omega(0)=0$. This modulus of continuity $\omega$ depends on $T$, $\Vert f\Vert_{L^

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.3
  • Proposition
  • proof
  • Lemma 2.1
  • Lemma 2.2
  • ...and 21 more