Uniform regularity for the Navier-Stokes equation with Navier boundary condition

TL;DR

This paper establishes uniform regularity results for the Navier-Stokes equations with Navier boundary conditions, enabling a rigorous derivation of the Euler system in the vanishing viscosity limit.

Contribution

It proves the existence of a uniform time interval with strong solutions bounded in conormal Sobolev spaces, facilitating the vanishing viscosity limit to Euler.

Findings

Uniform bounds in conormal Sobolev spaces

Single normal derivative bounded in L-infinity

Vanishing viscosity limit to Euler established

Abstract

We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier-Stokes equation with Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in $L^\infty$. This allows to get the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.