Asymptotic properties of the plane shear thickening fluids with bounded energy integral

TL;DR

This paper studies the long-term behavior of plane shear thickening fluids around obstacles, using advanced mathematical techniques to handle nonlinearities and establish decay estimates.

Contribution

It introduces a novel approach combining point-wise behavior and Brezis-Gallouet inequalities to analyze asymptotic decay in shear thickening fluids.

Findings

Decay estimates for velocity around obstacles

Handling of high-order nonlinear terms

Extension of techniques beyond Navier-Stokes equations

Abstract

In this note we investigate the asymptotic behavior of plane shear thickening fluids around a bounded obstacle. Different from the Navier-Stokes case considered by Gilbarg-Weinberger in \cite{GW1978}, where the good structure of the vorticity can be exploited and weighted energy estimates can be applied, we have to overcome the nonlinear term of high order. The decay estimates of the velocity was obtained by combining Point-wise Behavior Theorem in \cite{Galdi} and Brezis-Gallouet inequality in \cite{BG1980} together, which is independent of interest.