Stokes and Navier-Stokes equations with Navier boundary condition

TL;DR

This paper investigates the existence, uniqueness, and behavior of solutions to the stationary Stokes and Navier-Stokes equations with Navier boundary conditions, providing uniform estimates and analyzing the limit as the friction coefficient grows large.

Contribution

It establishes existence and uniqueness of solutions with minimal regularity assumptions and derives uniform estimates to study the asymptotic behavior as the friction coefficient tends to infinity.

Findings

Proved existence and uniqueness of solutions in various Sobolev spaces.

Derived uniform estimates independent of the friction coefficient.

Analyzed the limiting behavior of solutions as the friction coefficient approaches infinity.

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 $\alpha \rightarrow \infty$.