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 in the plane with a C^2-boundary. Navier boundary conditions can be expressed in the form w = (2 K - A) v . T and v . n = 0 on the boundary, where v is the velocity, w the vorticity, n a unit normal vector, T a unit tangent vector, and A is a bounded measurable function on the boundary. Such solutions have been considered for simply connected domains by Clopeau, Mikelić, and Robert under the assumption that the initial vorticity is bounded, and by Lopes Filho, Nussenzveig Lopes, and Planas under the assumption that the initial vorticity lies in L^p for some p > 2. We extend the results of these authors to non-simply connected domains. Assuming a particular bound on the growth of the L^p-norms of the initial vorticity with p, and also assuming that the boundary and the function A have fractionally greater smoothness, we obtain a bound on the rate of convergence in L^2 uniform over any finite time interval to the solution of the Euler equations in the vanishing viscosity limit. We also show that if the initial velocity is in H^3 and the boundary is C^3, then solutions to the Navier-Stokes equations with Navier boundary conditions converge in L^2 uniformly over any finite time interval to the solution to the Navier-Stokes equations with the usual no-slip boundary conditions as we let the function A grow large uniformly on the boundary.