Navier--Stokes Equations with Navier Boundary Conditions for a Bounded Domain in the Plane

Abstract

We consider solutions to the Navier--Stokes equations with Navier boundary conditions in a bounded domain $\Omega$ in ${\ensuremath{\BB{R}}}^2$ with a $C^2$-boundary $\Gamma$. Navier boundary conditions can be expressed in the form $\omega(v) = (2 \kappa - {\ensuremath{\alpha}}) v \cdot {\mbox{\boldmath $\tau$}}$ and $v \cdot \mathbf{n} = 0$ on $\Gamma$, where v is the velocity, $\omega(v)$ the vorticity, $\mathbf{n}$ a unit normal vector, ${\mbox{\boldmath $\tau$}}$ a unit tangent vector, and ${\ensuremath{\alpha}}$ is in $L^{\ensuremath{\infty}}(\Gamma)$. These boundary conditions were studied in the special case where ${\ensuremath{\alpha}} = 2 \kappa$ by J.-L. Lions and P.-L. Lions. We establish the existence, uniqueness, and regularity of such solutions, extending the work of Clopeau, Mikeli{ć, and Robert and of Lopes Filho, Nussenzveig Lopes, and Planas, which was restricted to simply connected domains and nonnegative ${\ensuremath{\alpha}}$. Assuming a particular bound on the growth of the $L^p$-norms of the initial vorticity with p (\textit{Yudovich vorticity}), and also assuming additional smoothness on $\Gamma$ and ${\ensuremath{\alpha}}$, we obtain a uniform-in-time bound on the rate of convergence in $L^2(\Omega)$ of solutions to the Navier--Stokes equations with Navier boundary conditions to the solution to the Euler equations in the vanishing viscosity limit. We also show that for smoother initial velocities, the solutions to the Navier--Stokes equations with Navier boundary conditions converge uniformly in time in $L^2(\Omega)$, and $L^2$ in time in $\dot{H}^1(\Omega)$, to the solution to the Navier--Stokes equations with the usual no-slip boundary conditions as we let ${\ensuremath{\alpha}}$ grow large uniformly on the boundary.