A fully non-linear version of the Euler incompressible equations: the semi-geostrophic system

TL;DR

This work analyzes the semi-geostrophic (SG) equations as a fully non-linear, dual-space reformulation of incompressible Euler, addressing existence, stability, and convergence to the 2-D Euler equations, including both measure-valued and smooth solutions. It introduces a new weak measure-solution framework that handles singular densities by reformulating the ill-defined product ρ ∇Ψ[ρ] via the generalized gradient (∇Ψ[ρ])^⊥ and the center of mass of ∂Ψ[ρ], together with Brenier’s polar factorization X = ∇Φ[ρ] ∘ g and Monge–Ampère relations det D^2 Ψ = ρ, det D^2 Φ = 1. Key contributions include global existence and stability of measure-valued solutions (and extension to bounded measures), local and, under regularity, global smooth solutions, Hölder-uniqueness via optimal transport, and convergence in 2-D to the incompressible Euler equation in a quasi-neutral limit, with proofs based on a modulated energy method and a regularity/expansion approach for smooth solutions. The paper also develops a transport–elliptic framework to prove uniqueness for 2-D Euler with bounded vorticity, and provides convergence results for both weak and strong SG toward EI under small-data scaling, supported by energy estimates along Wasserstein geodesics and epsilon-expansion arguments.

Abstract

This work gathers new results concerning the semi-geostrophic equations: existence and stability of measure valued solutions, existence and uniqueness of solutions under certain continuity conditions for the density, convergence to the incompressible Euler equations. Meanwhile, a general technique to prove uniqueness of sufficiently smooth solutions to non-linearly coupled system is introduced, using optimal transportation.

Theorems & Definitions (21)

• Theorem 1.1: Brenier, Br1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5: Brenier, Br1
• Proposition 2.6
• Theorem 3.1: Wang, W
• Corollary 3.2