Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On Liouville type theorem for a generalized stationary Navier-Stokes equations (1811.09051v2)

Published 22 Nov 2018 in math.AP

Abstract: In this paper we prove a Liouville type theorem for generalized stationary Navier-Stokes systems in $\Bbb R3$, which model non-Newtonian fluids, where the Laplacian term $\Delta u$ is replaced by the corresponding non linear operator $\bA_p( u)=\nabla \cdot ( |\bD(u)|{p-2} \bD(u))$ with $ \bD(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 $u\in W{1, p}(\Bbb R3)$ 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 impose the condition for the Liouville type theorem in terms of a potential function: if there exists a matrix valued potential function $\bV$ such that $ \nabla \cdot \bV =u$, whose $L{\frac{3p}{2p-3}} $ mean oscillation has the following growth condition at infinity, $$ \intmw_{B(r)} |\bV- \bV_{ B(r)} |{\frac{3p}{2p-3}} dx \le C r{\frac{9-4p}{2p-3}}\quad \forall 1< r< +\infty, $$ then $u\equiv 0$. In the case of the Navier-Stokes equations, $p=2$, this improves the previous results in the literature.

Summary

We haven't generated a summary for this paper yet.