Well-posedness of the hydrostatic Navier-Stokes equations

TL;DR

This paper proves local well-posedness of the hydrostatic Navier-Stokes equations for convex initial data using Gevrey regularity, expanding understanding of these equations' mathematical properties.

Contribution

It demonstrates local well-posedness under Gevrey regularity for convex initial data, relaxing the need for real-analyticity.

Findings

Local well-posedness established for convex initial data

Gevrey regularity suffices for well-posedness

Extends mathematical understanding of 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.