The Semigeostrophic--Euler Limit via Perturbative Monge--Ampère Estimates

We study the two-dimensional semigeostrophic system on the flat torus in the small-amplitude regime. We formulate the rescaled dynamics as the Lie–Poisson flow of a renormalized optimal-transport energy and expand this Hamiltonian in $C^1$. The leading term is the Euler Hamiltonian, while the first correction is an explicit cubic Monge–Ampère functional. We then derive quantitative consequences for the semigeostrophic–Euler limit: a perturbative scale-uniform endpoint Monge–Ampère estimate under Hessian pinching, an explicit logarithmic perturbative lifespan for the strong branch, fixed-slow-time $O(\text{eps})$ velocity convergence for canonically prepared strong branches, a conditional weak–strong rate-transfer corollary, and an $O({\text{eps}}^2)$ Wasserstein comparison for the physical densities.