Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A duality method for mean-field limits with singular interactions

Didier Bresch, Mitia Duerinckx, Pierre-Emmanuel Jabin

TL;DR

This work develops a duality-based framework to derive mean-field limits for both first- and second-order particle systems with singular interactions, allowing arbitrary square-integrable kernels at possibly vanishing temperature. By reformulating the problem via a backward dual Liouville equation, the authors study linear dual correlations in a weighted Hilbert space $L^2_f$, derive a BBGKY-type limit hierarchy for the dual correlations, and prove propagation of chaos toward the Vlasov equation $\partial_t f + v\cdot\nabla_x f + (K\!* f)\cdot\nabla_v f=\alpha\Delta_v f$. They further provide explicit hierarchical equations, robust a priori estimates, a truncated rescaled hierarchy with vanishing remainders, and a quantitative rate of convergence $\|F_{N,k}(t)-f(t)^{\otimes k}\|_{C_c^*} \lesssim (N^{-1/2}+N^{s(1/p-1/2)})^{e^{-Ct}}$, highlighting the role of regularity and diffusion. The method yields convergence to the 2d Euler/Navier–Stokes equations in the first-order setting and delivers convergence rates beyond previous approaches, with potential applicability to a broad class of singular kernels.

Abstract

We introduce a new approach to derive mean-field limits for first- and second-order particle systems with singular interactions. It is based on a duality approach combined with the analysis of linearized dual correlations, and it allows to cover for the first time arbitrary square-integrable interaction forces at possibly vanishing temperature. In case of first-order systems, it allows to recover in particular the mean-field limit to the 2d Euler and Navier-Stokes equations. The approach also provides convergence rates.

A duality method for mean-field limits with singular interactions

TL;DR

This work develops a duality-based framework to derive mean-field limits for both first- and second-order particle systems with singular interactions, allowing arbitrary square-integrable kernels at possibly vanishing temperature. By reformulating the problem via a backward dual Liouville equation, the authors study linear dual correlations in a weighted Hilbert space , derive a BBGKY-type limit hierarchy for the dual correlations, and prove propagation of chaos toward the Vlasov equation . They further provide explicit hierarchical equations, robust a priori estimates, a truncated rescaled hierarchy with vanishing remainders, and a quantitative rate of convergence , highlighting the role of regularity and diffusion. The method yields convergence to the 2d Euler/Navier–Stokes equations in the first-order setting and delivers convergence rates beyond previous approaches, with potential applicability to a broad class of singular kernels.

Abstract

We introduce a new approach to derive mean-field limits for first- and second-order particle systems with singular interactions. It is based on a duality approach combined with the analysis of linearized dual correlations, and it allows to cover for the first time arbitrary square-integrable interaction forces at possibly vanishing temperature. In case of first-order systems, it allows to recover in particular the mean-field limit to the 2d Euler and Navier-Stokes equations. The approach also provides convergence rates.
Paper Structure (9 sections, 15 theorems, 166 equations)

This paper contains 9 sections, 15 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Dual reformulation for mean field
  3. Dual (linear) correlations
  4. Limit hierarchy for dual correlations
  5. Quantitative estimates
  6. Explicit hierarchy for correlations
  7. A priori estimates on correlations
  8. Truncated rescaled hierarchy
  9. Proof of Theorem \ref{['theor:quant']}

Key Result

Theorem 1

Let $0\le\alpha<\infty$, let $K\in L^2_{loc}(\Omega;{\mathbb{R}}^d)$, and assume for convenience $K\in L^\infty_{loc}(|x|>1)$. Consider a global weak duality solution $F_N\in L^\infty_{loc}({\mathbb{R}}^+;L^1({\mathbb{D}}^N)\cap L^\infty({\mathbb{D}}^N))$ of the Liouville equation eq:Liouville, in t Then the following propagation of chaos holds: for all $k\ge1$, the $k$th marginal $F_{N,k}$, defin

Theorems & Definitions (29)

  • Theorem 1: Square-integrable interactions
  • Theorem 2: Error estimates
  • Theorem 3: First-order dynamics
  • Proposition 4
  • proof
  • Corollary 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 19 more