From second grade fluids to the Navier-Stokes equations

We consider the limit α→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 u0. If the product ‖u0‖L2‖u0‖H1 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/‖u0‖H14.