On the H ^S Theory of Hydrostatic Euler Equations

In this paper we study the two-dimensional hydrostatic Euler equations in a periodic channel. We prove the local existence and uniqueness of H ^S solutions under the local Rayleigh condition. This extends Brenier's (Nonlinearity 12(3):495-512, 1999) existence result by removing an artificial condition and proving uniqueness. In addition, we prove weak-strong uniqueness, mathematical justification of the formal derivation and stability of the hydrostatic Euler equations. These results are based on weighted H ^S a priori estimates, which come from a new type of nonlinear cancellation between velocity and vorticity.