The Semigeostrophic--Euler Limit via Perturbative Monge--Amp\`ere 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--Amp\`ere functional. We then derive quantitative consequences for the semigeostrophic--Euler limit: a perturbative scale-uniform endpoint Monge--Amp\`ere estimate under Hessian pinching, an explicit logarithmic perturbative lifespan for the strong branch, fixed-slow-time \(O(\eps)\) velocity convergence for canonically prepared strong branches, a conditional weak--strong rate-transfer corollary, and an \(O(\eps^2)\) Wasserstein comparison for the physical densities.