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Mean-field derivation of Landau-like equations

José Antonio Carrillo, Shuchen Guo, Pierre-Emmanuel Jabin

TL;DR

The paper establishes a rigorous mean-field derivation of a class of space-homogeneous Landau-like equations from stochastic particle systems on the torus. Using a relative-entropy framework, it derives a quantitative bound $H_N(t) \le (H_N(0) + C_1/N) e^{C_2 t}$ between the $N$-particle Liouville solution and the tensorized limit, which in turn yields a rate $\|f_{k,N}-f^{\otimes k}\|_{L^{\infty}([0,T],L^1(\mathbb{T}^{dk}))} \le C_3/\sqrt{N}$ for the $k$-particle marginals. This work extends propagation-of-chaos results to Landau-like dynamics via entropy methods, providing explicit, scalable convergence rates. The approach hinges on ellipticity of $a$, the structure $b=\nabla\cdot a$, and a cancellation lemma to control nonlinear interaction terms, with Grönwall-type arguments giving the final quantitative bounds.

Abstract

We derive a class of space homogeneous Landau-like equations from stochastic interacting particles. Through the use of relative entropy, we obtain quantitative bounds on the distance between the solution of the N-particle Liouville equation and the tensorised solution of the limiting Landau-like equation.

Mean-field derivation of Landau-like equations

TL;DR

The paper establishes a rigorous mean-field derivation of a class of space-homogeneous Landau-like equations from stochastic particle systems on the torus. Using a relative-entropy framework, it derives a quantitative bound between the -particle Liouville solution and the tensorized limit, which in turn yields a rate for the -particle marginals. This work extends propagation-of-chaos results to Landau-like dynamics via entropy methods, providing explicit, scalable convergence rates. The approach hinges on ellipticity of , the structure , and a cancellation lemma to control nonlinear interaction terms, with Grönwall-type arguments giving the final quantitative bounds.

Abstract

We derive a class of space homogeneous Landau-like equations from stochastic interacting particles. Through the use of relative entropy, we obtain quantitative bounds on the distance between the solution of the N-particle Liouville equation and the tensorised solution of the limiting Landau-like equation.
Paper Structure (3 sections, 4 theorems, 28 equations)

This paper contains 3 sections, 4 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof of Theorem \ref{['poc']}

Key Result

Theorem 2.2

Under assumptions above, there exists some positive constant $C_1$ and $C_2$ independent with $N$ such that the relative entropy of $f_N$ and $\bar{f}_N$ on the torus $\mathbb{T}^{dN}$ has the following estimate

Theorems & Definitions (5)

  • Definition 2.1: Entropy solution
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 3.1
  • Lemma 3.2