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arXiv · 2601.04622

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

Jingwen Han·Han Li
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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.

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Jingwen Han, Han Li. 2026-01-08. Liouville-type theorems for the stationary non-Newtonian fluids in a slab. https://arxiv.org/abs/2601.04622

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