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A geometric approximation to the Euler equations: the Vlasov-Monge-Ampere system

Yann Brenier, Gregoire Loeper

TL;DR

The paper studies the Vlasov-Monge-Ampère (VMA) system, a fully non-linear version of the Vlasov–Poisson system where the Poisson equation is replaced by a Monge–Ampère constraint det(I+ε^2 D^2 φ)=ρ, and frames it as a geometric approximation of the incompressible Euler equations in the spirit of Arnold–Ebin. It builds a penalty/approximate-geodesic formulation with a small parameter ε, employing weak Monge–Ampère theory and Brenier’s polar decomposition to connect VMA to Euler as an approximate geodesic on the group of volume-preserving maps. The main results include global existence of energy-preserving weak solutions, renormalization and well-posedness of characteristics, as well as local-in-time strong solutions in a periodic setting; it also establishes convergence to incompressible Euler for well-prepared data as ε→0 and clarifies the relation to Euler–Poisson. The work thus provides a rigorous kinetic-geometry framework linking VMA to VP and Euler and offers a systematic asymptotic and energy-based analysis using Kantorovich duality and polar-factorization tools.

Abstract

This paper studies the Vlasov-Monge-Ampere system (VMA), a fully non-linear version of the Vlasov-Poisson system (VP) where the (real) Monge-Ampere equation substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.

A geometric approximation to the Euler equations: the Vlasov-Monge-Ampere system

TL;DR

The paper studies the Vlasov-Monge-Ampère (VMA) system, a fully non-linear version of the Vlasov–Poisson system where the Poisson equation is replaced by a Monge–Ampère constraint det(I+ε^2 D^2 φ)=ρ, and frames it as a geometric approximation of the incompressible Euler equations in the spirit of Arnold–Ebin. It builds a penalty/approximate-geodesic formulation with a small parameter ε, employing weak Monge–Ampère theory and Brenier’s polar decomposition to connect VMA to Euler as an approximate geodesic on the group of volume-preserving maps. The main results include global existence of energy-preserving weak solutions, renormalization and well-posedness of characteristics, as well as local-in-time strong solutions in a periodic setting; it also establishes convergence to incompressible Euler for well-prepared data as ε→0 and clarifies the relation to Euler–Poisson. The work thus provides a rigorous kinetic-geometry framework linking VMA to VP and Euler and offers a systematic asymptotic and energy-based analysis using Kantorovich duality and polar-factorization tools.

Abstract

This paper studies the Vlasov-Monge-Ampere system (VMA), a fully non-linear version of the Vlasov-Poisson system (VP) where the (real) Monge-Ampere equation substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.

Paper Structure

This paper contains 19 sections, 24 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. The geometric origin of the Vlasov-Monge-Ampère system
  3. The Euler equations
  4. Approximate geodesics
  5. The polar decomposition Theorem
  6. The Vlasov-Monge-Ampère system
  7. Existence of global renormalized weak solutions
  8. Construction of a sequence of approximate solutions
  9. Conservation of energy
  10. Renormalized solutions and existence of characteristics
  11. Strong solutions
  12. The periodic Vlasov-Monge-Ampère system
  13. Existence of local strong solutions
  14. Construction of approximate solutions
  15. Asymptotic analysis
  16. ...and 4 more sections

Key Result

Theorem 2.2

Let \Omega be a bounded convex open subset of \mathbb R^d, let X\in L^2(\Omega;\mathbb R^d) and \rho_X=X\#dx, where dx is the Lebesgue measure on \Omega. Assume \rho_X to be a Lebesgue integrable function, or, equivalently, X to satisfy the non-degeneracy condition: Then there exists a unique pair (\nabla\Phi_X, \pi_X) where \Phi_X is a convex function and \pi_X\in S(\Omega), such that In this '

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Theorem 3.5
  • ...and 18 more