Upper and lower bounds of convergence rates for strong solutions of the generalized Newtonian fluids with non-standard growth conditions

TL;DR

This paper establishes precise decay rates for the difference between strong solutions of electrorheological fluid equations with variable power-law index, using heat equation asymptotics and Fourier splitting methods, improving previous results.

Contribution

It provides new bounds on the convergence rates of solutions for non-Newtonian fluids with variable growth conditions, extending prior work with improved decay estimates.

Findings

Decay rate of solution difference is proportional to (1+t)^(-γ/2).

Solution behavior is well approximated by the linear heat equation at large times.

Method improves previous results for fluids with constant power-law index.

Abstract

We consider the motion of an incompressible shear-thickening power-law-like non-Newtonian fluid in $R^3$ with a variable power-law index. This system of nonlinear partial differential equations arises in mathematical models of electrorheological fluids. The aim of this paper is to investigate the large-time behaviour of the difference $u-\tilde{u}$ where $u$ is a strong solution of the given equations with the initial data $u_0$ and $\tilde{u}$ is the strong solution of the same equations with perturbed initial data $u_0+w_0$. The initial perturbation $w_0$ is not required to be small, but is assumed to satisfy certain decay condition. In particular, we can show that $(1+t)^{-\frac{\gamma}{2}}\lesssim \|u(t)-\tilde{u}(t)\|_2\lesssim(1+t)^{-\frac{\gamma}{2}}$, for sufficiently large $t>0$, where $\gamma\in(2,\frac{5}{2})$. The proof is based on the observation that the solution of the linear heat equation describes the asymptotic behaviour of the solutions of the electrorheological fluids well for sufficiently large time $t>0$, and the generalized Fourier splitting method with an iterative argument. Furthermore, it will also be discussed that the argument used in the present paper can improve the previous results for the generalized Newtonian fluids with a constant power-law index.