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Contractivity of Wasserstein distance and exponential decay for the Landau equation with Maxwellian molecules

F. -U. Caja-Lopez, M. G. Delgadino, M. -P. Gualdani, M. Taskovic

TL;DR

This work analyzes monotonicity properties for the space-homogeneous Landau equation, focusing on Maxwellian molecules and soft potentials. It proves exponential decay of the relative $L^2$ distance to equilibrium for Maxwell molecules and establishes contractivity of the entropic 2-Wasserstein distance via two independent approaches, including a duality-based dissipation formula. It further extends contractivity to soft potentials, providing explicit rates under suitable $L^p$ or $L^\infty$ bounds and linking these results to known Fournier–Guerin uniqueness results. The paper thus connects Fisher-information monotonicity to decay of other physically meaningful functionals and offers new, explicit expressions for Wasserstein dissipation, highlighting a gradient-flow structure in the Landau dynamics with Maxwellian and soft-potentials interactions.

Abstract

Following the breakthrough work of Guillen and Silvestre \cite{GS24}, that shows that the Fisher information is monotonically decreasing for solutions to the homogeneous Landau equation, we study, for the same equation, the monotonicity properties of other physically relevant functionals. In the case of Maxwellian molecules, we show that the relative $L^2$ norm with respect to the equilibrium decays exponentially fast in time and is monotonically decreasing after some time. Moreover, still for the Maxwellian case, we provide a novel and short quantitative proof of time monotonicity of the entropic Wasserstein metric. For soft potentials, we show that the Wasserstein metric is contractive, conditional to $L^1(0,T,L^p(\mathbb{R}^3))$ bound for the solution. This result provides an alternative proof of the Fournier and Fournier-Guerin uniqueness theorem in \cite{fournier2009well_posedness_soft_potentials} \cite{fournier2010uniqueness_Coulomb}

Contractivity of Wasserstein distance and exponential decay for the Landau equation with Maxwellian molecules

TL;DR

This work analyzes monotonicity properties for the space-homogeneous Landau equation, focusing on Maxwellian molecules and soft potentials. It proves exponential decay of the relative distance to equilibrium for Maxwell molecules and establishes contractivity of the entropic 2-Wasserstein distance via two independent approaches, including a duality-based dissipation formula. It further extends contractivity to soft potentials, providing explicit rates under suitable or bounds and linking these results to known Fournier–Guerin uniqueness results. The paper thus connects Fisher-information monotonicity to decay of other physically meaningful functionals and offers new, explicit expressions for Wasserstein dissipation, highlighting a gradient-flow structure in the Landau dynamics with Maxwellian and soft-potentials interactions.

Abstract

Following the breakthrough work of Guillen and Silvestre \cite{GS24}, that shows that the Fisher information is monotonically decreasing for solutions to the homogeneous Landau equation, we study, for the same equation, the monotonicity properties of other physically relevant functionals. In the case of Maxwellian molecules, we show that the relative norm with respect to the equilibrium decays exponentially fast in time and is monotonically decreasing after some time. Moreover, still for the Maxwellian case, we provide a novel and short quantitative proof of time monotonicity of the entropic Wasserstein metric. For soft potentials, we show that the Wasserstein metric is contractive, conditional to bound for the solution. This result provides an alternative proof of the Fournier and Fournier-Guerin uniqueness theorem in \cite{fournier2009well_posedness_soft_potentials} \cite{fournier2010uniqueness_Coulomb}

Paper Structure

This paper contains 6 sections, 12 theorems, 163 equations, 1 table.

Table of Contents

  1. Introduction
  2. The collision operator for Maxwellian molecules
  3. Exponential decay of relative $L^2$ norm
  4. Contractivity of Entropic 2-Wasserstein distance
  5. Proof of Theorem \ref{['thm:w2_dissipation']}: second method
  6. Proof of Theorem \ref{['thm:w2_estimate_non_Coulomb']}

Key Result

Theorem 1

Let $f$ be a solution to (L) with $\alpha \equiv 1$ and initial data $f_{in}$ such that $\int f_{in} (1, v, |v|^2) \;dv = (1,0,d)$. Let $m$ be the corresponding unique equilibrium solution, $m(v) :=(2\pi)^{-\frac{d}{2}}e^{-\frac{1}{2}\vert{v}\vert^{2}}$. Then the relative $L^2$-norm decays exponentially fast in time, where $c_0$ depends only on the dimension $d$. Moreover, $E(t)$ is monotonicall

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 15 more