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

Jingwen Han, Han Li

TL;DR

This work proves Liouville-type theorems for stationary shear thickening non-Newtonian fluids in a slab, showing that under mild growth or boundedness conditions, nontrivial solutions cannot persist. The authors develop Saint-Venant-type energy estimates for the local Dirichlet integral, and employ refined Korn-type inequalities and Bogovskii lifts to control nonlinear advection and pressure terms. They establish precise decay/growth criteria for axisymmetric solutions and extend these to general flows, demonstrating that small or decaying energy in large slabs forces the velocity field to vanish. The results generalize corresponding Navier–Stokes Liouville-type theorems to the non-Newtonian, shear-thickening regime in a slab, providing a rigorous description of far-field behavior and rigidity of stationary states in this class of fluids.

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.

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

TL;DR

This work proves Liouville-type theorems for stationary shear thickening non-Newtonian fluids in a slab, showing that under mild growth or boundedness conditions, nontrivial solutions cannot persist. The authors develop Saint-Venant-type energy estimates for the local Dirichlet integral, and employ refined Korn-type inequalities and Bogovskii lifts to control nonlinear advection and pressure terms. They establish precise decay/growth criteria for axisymmetric solutions and extend these to general flows, demonstrating that small or decaying energy in large slabs forces the velocity field to vanish. The results generalize corresponding Navier–Stokes Liouville-type theorems to the non-Newtonian, shear-thickening regime in a slab, providing a rigorous description of far-field behavior and rigidity of stationary states in this class of fluids.

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 -norm grows mildly as the radius grows. Also, a bounded general solution must be trivial if 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.
Paper Structure (4 sections, 10 theorems, 154 equations)

This paper contains 4 sections, 10 theorems, 154 equations.

Table of Contents

  1. Introduction and Main Results
  2. PRELIMINARIES
  3. Liouville-type Theorem for Axisymmetric Solutions
  4. GENERAL NON-NEWTONIAN FLUIDS IN A SLAB

Key Result

Theorem 1.1

Assume that $2\leq p<+\infty$ and $\Omega= \mathbb{R}^2\times (0, 1)$. Let ${\boldsymbol{u}}$ be the weak solution of the equations eqsteadynNfs in a slab with no-slip boundary conditions noslipboun, then ${\boldsymbol{u}}\equiv 0$, provided that the Dirichlet integral is finite, i.e.,

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Korn's inequality in the cylindrical domains
  • ...and 11 more