We consider solutions to the Navier-Stokes equations with Navier boundary conditions in a bounded domain Ω in R 2 with a C 2boundary 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 p -norms of the initial vorticity with p, and also assuming that for some ǫ > 0, Γ is C 2,1/2+ǫ and α is in H 1/2+ǫ (Γ) + C 1/2+ǫ (Γ), we obtain a bound on the rate of convergence in L ∞ ([0, T ]; L 2 (Ω) ∩ L 2 (Γ)) to the solution to the Euler equations in the vanishing viscosity limit. We also show that if the initial velocity is in H 3 (Ω) and Γ is C 3 , then solutions to the Navier-Stokes equations with Navier boundary conditions converge in L ∞ ([0, T ]; L 2 (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. 1991 Mathematics Subject Classification. Primary 76D05, 76C99.

Let u be a solution to the Navier-Stokes equations with viscosity ν in a bounded domain Ω in R d , d ≥ 2, and let ū be the solution to the Euler equations in Ω. In 1983 Tosio Kato showed that for sufficiently regular solutions,being a layer of thickness cν near the boundary. We show that Kato's condition is equivalent to ν ω(u) 2 X → 0 as ν → 0, where ω(u) is the vorticity (curl) of u, and is also equivalent to ν −1 u 2 X → 0 as ν → 0.

We prove existence and uniqueness of a weak solution to the incompressible 2D Euler equations in the exterior of a bounded smooth obstacle when the initial data is a bounded divergence-free velocity field having bounded scalar curl. This work completes and extends the ideas outlined by P. Serfati for the same problem in the whole-plane case. With non-decaying vorticity, the Biot-Savart integral does not converge, and thus velocity cannot be reconstructed from vorticity in a straightforward way. The key to circumventing this difficulty is the use of the Serfati identity, which is based on the Biot-Savart integral, but holds in more general settings.

We say that the vanishing viscosity limit holds in the classical sense if the velocity for a solution to the Navier-Stokes equations converges in the energy norm uniformly in time to the velocity for a solution to the Euler equations. We prove, for a bounded domain in dimension 2 or higher, that the vanishing viscosity limit holds in the classical sense if and only if a vortex sheet forms on the boundary.2000 Mathematics Subject Classification. Primary 76D05, 76B99, 76D99.

No abstract