Optimal Gevrey stability of hydrostatic approximation for the Navier-Stokes equations in a thin domain

TL;DR

This paper establishes optimal Gevrey stability for the hydrostatic approximation of the Navier-Stokes equations in a thin domain, demonstrating convergence and well-posedness under optimal regularity conditions.

Contribution

It proves the limit from anisotropic Navier-Stokes to hydrostatic systems with optimal Gevrey regularity and shows well-posedness in this space, improving previous results.

Findings

Justifies the hydrostatic limit for convex initial data with optimal Gevrey regularity.

Demonstrates well-posedness of the hydrostatic Navier-Stokes/Prandtl system in optimal Gevrey space.

Improves the Gevrey index from 9/8 to 3/2 in stability analysis.

Abstract

In this paper, we study the hydrostatic approximation for the Navier-Stokes system in a thin domain. When the convex initial data with Gevrey regularity of optimal index 3/2 in x variable and Sobolev regularity in y variable, we justify the limit from the anisotropic Navier-Stokes system to the hydrostatic Navier-Stokes/Prandtl system. Due to our method in the paper is independent of {\epsilon}, by the same argument, we also get the hydrostatic Navier-Stokes/Prandtl system is well-posedness in the optimal Gevrey space. Our results improve the Gevrey index in [14, 34] whose Gevrey index is 9/8 .