  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.
