A Liouville theorem for stationary incompressible fluids of von Mises type

TL;DR

This paper proves a Liouville theorem for certain stationary incompressible fluid flows, showing that bounded solutions are constant under specific conditions, extending previous results to fluids with von Mises-type stress-strain relations.

Contribution

It establishes a Liouville property for perfectly plastic fluids with linear growth dissipative potentials, generalizing known results to the case p=1.

Findings

Liouville theorem holds for p-fluids with p>1

Established Liouville property for Prandtl-Eyring fluids

Proved Liouville theorem for von Mises-type fluids with linear growth potentials

Abstract

We consider entire solutions $u$ of the equations describing the stationary flow of a generalized Newtonian fluid in 2D concentrating on the question, if a Liouville-type result holds in the sense that the boundedness of $u$ implies its constancy. A positive answer is true for $p$-fluids in the case $p>1$ (including the classical Navier-Stokes system for the choice $p=2$), and recently we established this Liouville property for the Prandtl-Eyring fluid model, for which the dissipative potential has nearly linear growth. Here we finally discuss the case of perfectly plastic fluids whose flow is governed by a von Mises-type stress-strain relation formally corresponding to the case $p=1$. it turns out that, for dissipative potentials of linear growth, the condition of $\mu$-ellipticity with exponent $\mu<2$ is sufficient for proving the Liouville theorem.