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We study the stationary Stokes and Navier-Stokes equations withnon-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 regularityon the friction coefficient $\alpha$. Moreover, we deduce uniform estimates onthe solution with respect to $\alpha$ which enables us to analyze the behaviorof the solution when $\alpha \rightarrow \infty$.

Stokes and Navier-Stokes equations with Navier boundary condition