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

We consider solutions to the Navier-Stokes equations with Navier boundary conditions in a bounded domain Ω in R with a Cboundary Γ. Navier boundary conditions can be expressed in the form ω(v) = (2κ−α)v · τ and v ·n = 0 on Γ, where v is the velocity, ω(v) the vorticity, n a unit normal vector, τ a unit tangent vector, and α is in L∞(Γ). Such solutions have been considered in [2] and [3], and, in the special case where α = 2κ, by J.L. Lions in [10] and by P.L. Lions in [11]. We extend the results of [2] and [3] to non-simply connected domains. Assuming, as Yudovich does in [15], a particular bound on the growth of the L-norms of the initial vorticity with p, and also assuming that for some ǫ > 0, Γ is C and α is in H(Γ) +C(Γ), we obtain a bound on the rate of convergence in L([0, T ];L(Ω) ∩ L(Γ)) to the solution to the Euler equations in the vanishing viscosity limit. We also show that if the initial velocity is in H(Ω) and Γ is C, then solutions to the Navier-Stokes equations with Navier boundary conditions converge in L∞([0, T ];L(Ω)) to the solution to the Navier-Stokes equations with the usual no-slip boundary conditions as we let α grow large uniformly on the boundary.