Asymptotic profiles for the second grade fluids equations in R²

In the present paper, we study the long time behaviour of the solutions of the second grade fluids equations in dimension 3. Using scaling variables and energy estimates in weighted Sobolev spaces, we describe the first order asymptotic profiles of these solutions. In particular, we show that the solutions of the second grade fluids equations converge to self-similar solutions of the heat equations, which are explicit and depend on the initial data. Since this phenomenon occurs also for the Navier-Stokes equations, it shows that the fluids of second grade behave asymptotically like Newtonian fluids.