Well-posedness of the hydrostatic Navier–Stokes equations

Abstract

We address the local well-posedness of the hydrostatic Navier–Stokes equations. These equations, sometimes called reduced Navier–Stokes/Prandtl, appear as a formal limit of the Navier–Stokes system in thin domains, under certain constraints on the aspect ratio and the Reynolds number. It is known that without any structural assumption on the initial data, real-analyticity is both necessary and sufficient for the local well-posedness of the system. In this paper we prove that for convex initial data, local well-posedness holds under simple Gevrey regularity.