On the convergence of the two-dimensional second grade fluid model to the Navier-Stokes equation

We consider the equations governing the motion of incompressible second grade fluids in a bounded two-dimensional domain with Navier-slip boundary conditions. We first prove that the corresponding solutions are uniformly bounded with respect to the normal stress modulus α in the L^∞-H¹ and the L²-H² time-space norms. Next, we study their asymptotic behavior when α tends to zero, prove that they converge to regular solutions of the Navier-Stokes equations and give the rate of convergence in terms of α.