Uniform bounds and the inviscid limit for the Navier-Stokes equations with Navier boundary conditions

TL;DR

This paper proves uniform bounds and the inviscid limit for Navier-Stokes equations with Navier boundary conditions, reducing regularity requirements and establishing new initial data classes for Euler equations in half-space settings.

Contribution

It lowers the regularity needed for the inviscid limit and introduces a new class of initial data for local existence and uniqueness of Euler equations.

Findings

Established inviscid limit under weaker regularity assumptions.

Proved local existence and uniqueness for Euler equations with conormal initial data.

Derived uniform bounds for solutions with Navier boundary conditions.

Abstract

We consider the vanishing viscosity problem for solutions of the Navier-Stokes equations with Navier boundary conditions in the half-space. We lower the currently known conormal regularity needed to establish that the inviscid limit holds. Our requirement for the Lipschitz initial data is that the first four conormal derivatives are bounded along with two for the gradient. In addition, we establish a new class of initial data for the local existence and uniqueness for the Euler equations in the half-space or a channel for initial data in the conormal space without conormal requirements on the gradient.