Temporal decay of strong solutions for generalized Newtonian fluids with variable power-law index

TL;DR

This paper studies the decay behavior of strong solutions for a generalized Newtonian fluid model with variable power-law index, revealing specific decay rates for the solution's norms using Fourier analysis.

Contribution

It establishes decay rates for strong solutions of a variable power-law Newtonian fluid model, including conditions for decay of derivatives, using Fourier splitting methods.

Findings

L^2-norm of solutions decays at rate (1 + t)^{-3/4}

Derivative of solutions decays at rate (1 + t)^{-5/4} under small initial H^1-norm

Provides mathematical analysis relevant to electrorheological fluid models

Abstract

We consider the motion of a power-law-like generalized Newtonian fluid in R^3, where the power-law index is a variable function. This system of nonlinear partial differential equations arises in mathematical models of electrorheological fluids. The aim of this paper is to investigate the decay properties of strong solutions for the model, based on the Fourier splitting method. We first prove that the L^2-norm of the solution has the decay rate (1 + t)^{-3/4}. If the H^1-norm of the initial data is sufficiently small, we further show that the derivative of the solution decays in L^2-norm at the rate (1 + t)^{-5/4}.