From second grade fluids to the Navier-Stokes equations

We consider the limit $\alpha\to0$ 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 $\|u_0\|_{L^2}\|u_0\|_{H^1}$ 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 $C/\|u_0\|_{H^1}^4$.