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

TL;DR

This paper establishes quantitative stability and lifespan bounds for the semigeostrophic system's approximation by Euler equations in two dimensions, providing new insights into their relationship and long-term behavior.

Contribution

It introduces novel $O( ext{epsilon})$ stability estimates and a lifespan lower bound with a logarithmic improvement for the semigeostrophic system in the small-amplitude regime.

Findings

Proves $O( ext{epsilon})$ velocity stability in $L^2$ norm.

Establishes $O( ext{epsilon})$ Wasserstein distance estimate for densities.

Provides a lifespan lower bound $T_*( ext{epsilon}) o ext{infinity}$ as epsilon $ o 0$.

Abstract

We study the two-dimensional semigeostrophic system on the flat torus in the small-amplitude scaling and quantify its approximation by incompressible Euler in dual variables. On a natural perturbative bootstrap window for the Monge--Amp\`ere coupling, we prove two strong stability results: an $O(\eps)$ estimate for the velocity in $L^2$, and an $O(\eps)$ estimate in Wasserstein distance for the associated physical densities. The latter is deduced from a more general comparison theorem, independent of the bootstrap regime, which combines the deterministic flow representation for the smooth Euler solution with a superposition representation for the semigeostrophic continuity equation. We also prove a lifespan lower bound with a logarithmic improvement over the standard hyperbolic scale, namely $T_*(\eps)\gtrsim \eps^{-1}\log\log(1/\eps)$ in physical time.