Inviscid limit on $L^p$-based Sobolev conormal spaces for the 3D Navier-Stokes equations with the Navier boundary conditions

TL;DR

This paper proves uniform bounds and the inviscid limit for solutions of 3D Navier-Stokes with Navier boundary conditions in Sobolev conormal spaces, extending previous results with weaker regularity assumptions.

Contribution

It extends vanishing viscosity results by weakening regularity assumptions and establishes existence and uniqueness of Euler solutions with bounded conormal derivatives.

Findings

Uniform bounds in Sobolev conormal spaces for Navier-Stokes solutions

Inviscid limit established under weaker regularity conditions

Existence and uniqueness of Euler solutions with bounded derivatives

Abstract

We establish uniform bounds and the inviscid limit in $L^p$-based Sobolev conormal spaces for the solutions of the Navier-Stokes equations with the Navier boundary conditions in the half-space. We extend the vanishing viscosity results of~\cite{BdVC1} and~\cite{AK1} by weakening the normal and the conormal regularity assumptions, respectively. We require the initial data to be Lipschitz with three integrable conormal derivatives. We also assume that the initial normal derivative has one or two integrable conormal derivative depending on the sign of the friction coefficient. Finally, we establish the existence and uniqueness of the Euler equations with a bounded normal derivate, two bounded conormal derivatives, and three integrable conormal derivatives.