From second grade fluids to the Navier-Stokes equations

We consider the limit $\alpha \to 0$ for a second grade fluid on a bounded domain with Dirichlet boundary conditions. We show convergence towards a solution of the Navier-Stokes equations under two different types of hypothesis on the initial velocity $u_0$. If the product $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^1}$ is sufficiently small we prove global-in-time convergence. If there is no smallness assumption we obtain local-in-time convergence up to the time $C\mathrm{/}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^1}^4$.