Asymptotic profiles for the second grade fluids equations in $\mathbb{R}^3$

In the present paper, we study the long time behaviour of the solutions of the second grade fluids equations in $\mathbb{R}^3$. 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 equation, 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.

Keywords

second grade fluids equations, asymptotic profiles, self-similar solutions

2010 Mathematics Subject Classification

35-xx, 76Xxx

Full Text (PDF format)

Published 24 June 2014