On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations

Abstract

In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in \(\mathbb R^3\), having the diffusion term \({\varvec{A}}_p( u)=\nabla \cdot ( |{\varvec{D}}(u)|^{p-2} {\varvec{D}}(u))\) with \( {\varvec{D}}(u) = \frac{1}{2} (\nabla u + (\nabla u)^{ \top })\), \(3/2<p< 3\). In the case \(3/2< p\le 9/5\), we show that a suitable weak solution \(u\in W^{1, p}(\mathbb R^3)\) satisfying \( \liminf _{R \rightarrow \infty } |u_{ B(R)}| =0\) is trivial, i.e., \(u\equiv 0\). On the other hand, for \(9/5<p<3\) we prove the following Liouville type theorem: if there exists a matrix valued function \({\varvec{V}}= \{V_{ ij}\}\) such that \( \partial _jV_{ ij} =u_i\)(summation convention), whose \(L^{\frac{3p}{2p-3}} \) mean oscillation has the following growth condition at infinity,

$$\begin{aligned} {\int \!\!\!\!\!\!-}_{B(r)} |{\varvec{V}}- {\varvec{V}}_{ B(r)} |^{\frac{3p}{2p-3}} \mathrm{d}x \le C r^{\frac{9-4p}{2p-3}}\quad \forall 1< r< +\infty , \end{aligned}$$

then \(u\equiv 0\).

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Acknowledgements

Chae was partially supported by NRF Grants 2016R1A2B3011647, while Wolf has been supported by NRF Grants 2017R1E1A1A01074536.

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Correspondence to Dongho Chae.

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Chae, D., Wolf, J. On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations. J Nonlinear Sci 30, 1503–1517 (2020). https://doi.org/10.1007/s00332-020-09615-y

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Keywords

  • Non-Newtonian fluid equations
  • Liouville type theorem

Mathematics Subject Classification

  • 35Q30
  • 76D05
  • 76D03