Liouville-type theorems for the stationary non-Newtonian fluids in a slab

TL;DR

This paper proves Liouville-type theorems for stationary shear thickening fluid equations in a slab, showing triviality of solutions under certain growth conditions, inspired by Navier-Stokes analysis.

Contribution

It establishes new Liouville theorems for shear thickening fluids, including growth conditions and Korn's inequality estimates, extending prior Navier-Stokes results.

Findings

Axisymmetric solutions are trivial under mild growth conditions.

Bounded solutions with bounded $ru^r$ are trivial.

Introduces a Saint-Venant type estimate for solution growth.

Abstract

In this paper, we investigate Liouville-type theorems for stationary solutions to the shear thickening fluid equations in a slab. We show that the axisymmetric solution must be trivial if its local $L^\infty$-norm grows mildly as the radius $R$ grows. Also, a bounded general solution $u$ must be trivial if $ru^r$ is bounded. The proof is inspired by the work of Bang, Gui, Wang, and Xie [J. Fluid Mech. 1005 (2025)] for the Navier-Stokes equations, and the key point is to establish a Saint-Venant type estimate that characterizes the growth of the local Dirichlet integral of nontrivial solutions. One new ingredient is the estimate of the constant in Korn's inequality over different domains.