Remarks on the inviscid limit for the Navier-Stokes equations for uniformly bounded velocity fields

TL;DR

This paper investigates the inviscid limit of the Navier-Stokes equations in a half space, establishing conditions under which solutions converge in energy norm when boundary behavior is controlled.

Contribution

It provides new sufficient conditions involving boundary regularity and boundedness for the inviscid limit to hold in the energy norm.

Findings

Inviscid limit holds in energy norm under equicontinuity conditions.

Uniform boundedness of tangential velocity and integrability of its gradient near boundary are sufficient.

Results apply to Navier-Stokes solutions with specific boundary regularity assumptions.

Abstract

We consider the vanishing viscosity limit of the Navier-Stokes equations in a half space, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the product of the components of the Navier-Stokes solutions are equicontinuous at $x_2=0$. A sufficient condition for this to hold is that the tangential Navier-Stokes velocity remains uniformly bounded and has a uniformly integrable tangential gradient near the boundary.