Dynamics of Partial Differential Equations

Volume 11 (2014)

Number 2

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

Pages: 125 – 165

DOI: http://dx.doi.org/10.4310/DPDE.2014.v11.n2.a2

Author

Olivier Coulaud (Department of Mathematics, Université Paris-Sud, Orsay, France)

Abstract

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

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