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

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.