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

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 $L^{\infty }((0,T)\times ø)$, where $T$ is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also.

45pages

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