Vanishing Viscous Limits for 3D Navier-Stokes Equations with A Navier-Slip Boundary Condition

TL;DR

This paper studies the behavior of 3D Navier-Stokes solutions with Navier-slip boundary conditions as viscosity vanishes, proving convergence to Euler solutions with explicit rates on smooth domains.

Contribution

It establishes higher order regularity estimates for boundary layers and proves convergence of Navier-Stokes solutions to Euler solutions with explicit rates, under Navier-slip conditions.

Findings

Convergence of Navier-Stokes to Euler solutions in specified norms.

Higher order regularity estimates for boundary layer solutions.

Explicit rates of convergence as viscosity tends to zero.

Abstract

In this paper, we investigate the vanishing viscosity limit for solutions to the Navier-Stokes equations with a Navier slip boundary condition on general compact and smooth domains in $\mathbf{R}^3$. We first obtain the higher order regularity estimates for the solutions to Prandtl's equation boundary layers. Furthermore, we prove that the strong solution to Navier-Stokes equations converges to the Eulerian one in $C([0,T];H^1(\Omega))$ and $L^\infty((0,T)\times\o)$, where $T$ is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also.