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The Semigeostrophic-Euler Limit: Lifespan Lower Bounds and $O(\varepsilon)$ Velocity Stability

Victor Armegioiu

TL;DR

This work analyzes the 2D semigeostrophic system in dual variables on the torus under a small-amplitude, slow-time scaling and rigorously quantifies its convergence toward incompressible Euler. By coupling incompressible transport with a perturbative Poisson–Monge–Ampère constraint and exploiting sharp elliptic estimates, the authors establish a lifespan bound with a log–log gain in slow time, yielding physical persistence $T_*(\varepsilon) \gtrsim \frac{1}{\varepsilon}\,|\log\log\varepsilon|$, and an $O(\varepsilon)$ velocity stability bound in $L^2$ on bootstrap windows. The analysis hinges on a flow-based stability framework, Wente-type control of the Hessian determinant, and endpoint Calderón–Zygmund bounds, providing a quantitative bridge from SG$^{\varepsilon}$ to Euler that is both longer-lived and more stable in velocity than prior weak convergence results. The results inform long-time approximations of geophysical flows by Euler and illuminate how nonlinear Monge–Ampère corrections remain perturbative in a carefully designed bootstrap regime.

Abstract

We study the two-dimensional semigeostrophic (SG$^{\varepsilon}$) system on the torus in the small-amplitude scaling and its convergence to incompressible Euler in the dual (geostrophic) formulation. Within a natural bootstrap regime for the Poisson/Monge-Ampère coupling, we obtain two main results. First, we prove a lifespan lower bound in slow time with a \emph{log-log} gain; in physical time this yields persistence at least on the scale $\varepsilon^{-1}|\log\log \varepsilon|$. Second, on any bootstrap window we establish a strong velocity-stability estimate with rate $O(\varepsilon)$ in $L^2$, complementing Loeper's $O(1/\varepsilon)$ existence time and $\varepsilon^{2/3}$ weak convergence rate. The proofs combine the incompressible transport structure with a sharp elliptic control of the velocity gradient and a flow-based stability argument. Overall, the results give a clean quantitative bridge from SG$^{\varepsilon}$ to Euler that is both longer-lived (by a log-log factor) and quantitatively stable in velocity.

The Semigeostrophic-Euler Limit: Lifespan Lower Bounds and $O(\varepsilon)$ Velocity Stability

TL;DR

This work analyzes the 2D semigeostrophic system in dual variables on the torus under a small-amplitude, slow-time scaling and rigorously quantifies its convergence toward incompressible Euler. By coupling incompressible transport with a perturbative Poisson–Monge–Ampère constraint and exploiting sharp elliptic estimates, the authors establish a lifespan bound with a log–log gain in slow time, yielding physical persistence , and an velocity stability bound in on bootstrap windows. The analysis hinges on a flow-based stability framework, Wente-type control of the Hessian determinant, and endpoint Calderón–Zygmund bounds, providing a quantitative bridge from SG to Euler that is both longer-lived and more stable in velocity than prior weak convergence results. The results inform long-time approximations of geophysical flows by Euler and illuminate how nonlinear Monge–Ampère corrections remain perturbative in a carefully designed bootstrap regime.

Abstract

We study the two-dimensional semigeostrophic (SG) system on the torus in the small-amplitude scaling and its convergence to incompressible Euler in the dual (geostrophic) formulation. Within a natural bootstrap regime for the Poisson/Monge-Ampère coupling, we obtain two main results. First, we prove a lifespan lower bound in slow time with a \emph{log-log} gain; in physical time this yields persistence at least on the scale . Second, on any bootstrap window we establish a strong velocity-stability estimate with rate in , complementing Loeper's existence time and weak convergence rate. The proofs combine the incompressible transport structure with a sharp elliptic control of the velocity gradient and a flow-based stability argument. Overall, the results give a clean quantitative bridge from SG to Euler that is both longer-lived (by a log-log factor) and quantitatively stable in velocity.
Paper Structure (21 sections, 25 theorems, 179 equations)

This paper contains 21 sections, 25 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Small-amplitude scaling and the SG-Euler limit
  3. Previous work on semigeostrophic dynamics
  4. Contribution of this paper
  5. Organisation of the paper
  6. Notation and setup
  7. Physical background and model scope
  8. Dual (geostrophic) variables and mass rearrangement.
  9. Polar factorization and the dual formulation on $\mathbb{T}^d$
  10. Transport structure and flows
  11. Main results
  12. Preliminaries
  13. How these ingredients interact.
  14. Takeaway for the paper.
  15. Proof of the Main Lifespan Theorem
  16. ...and 6 more sections

Key Result

Proposition 3.1

Let $\rho^0\in L^\infty(\mathbb{T}^2)$ be a nonnegative probability density and let $u_i=\nabla^\perp\psi_i$ be divergence-free vector fields with associated measure-preserving flows $X_i:\mathbb{T}^2\to\mathbb{T}^2$ such that $\rho_i:=(X_i)_{\#}\rho^0\in L^\infty(\mathbb{T}^2)$, $i=1,2$. Assume the Then the following holds

Theorems & Definitions (51)

  • Proposition 3.1
  • proof
  • Theorem 4.1: Bootstrap lifespan with log-log gain
  • proof : Proof Strategy
  • Theorem 4.2: $O(\varepsilon)$ velocity stability on the bootstrap window
  • proof : Proof Strategy
  • Definition 5.1: Bootstrap Condition
  • Proposition 5.2
  • proof
  • Theorem 6.1: Lifespan Lower Bound
  • ...and 41 more