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∞L∞-H1H1 and the L2L2-H2H2 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.