Asymptotic profiles for the second grade fluids equations in R^2

TL;DR

This paper investigates the long-term behavior of second grade fluid solutions in three dimensions, demonstrating their convergence to heat equation self-similar solutions, thus revealing their asymptotic similarity to Newtonian fluids.

Contribution

It provides the first detailed analysis of the asymptotic profiles of second grade fluids in 3D using energy estimates and scaling techniques.

Findings

Solutions converge to heat equation self-similar profiles

Second grade fluids behave asymptotically like Newtonian fluids

Explicit asymptotic profiles depend on initial data

Abstract

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.