Stokes and Navier-Stokes equations with Navier boundary condition

Abstract

We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $\Omega \subset {\mathbb{R}}^3$ of class ${\mathcal{C}}^{1,1}$. We prove existence, uniqueness of weak and strong solutions in ${\mathbf{W}}^{1,p}(\Omega )$ and ${\mathbf{W}}^{2,p}(\Omega )$ for all $1<p<\infty$ considering minimal regularity on the friction coefficient $\alpha$. Moreover, we deduce uniform estimates on the solution with respect to $\alpha$ which enables us to analyze the behavior of the solution when $\alpha \to \infty$.

corrected proof of Theorem 6.3