The Liouville type theorems for the steady Navier-Stokes equations and the self-similar Euler equations on $\Bbb R^3$

Details The Liouville type theorems for the steady Navier-Stokes equations and the self-similar Euler equations on $\Bbb R^3$ The Liouville type theorems for the steady Navier-Stokes equations and the self-similar Euler equations on $\Bbb R^3$

2011-05-18

arxiv.org/abs/1105.3639v4

Mathematics

Analysis of PDEs

14 pages

Scientific paper

In this paper we study the Liouville type properties for the steady Navier-Stokes equations(NS) and the self-similar Euler equations. For the smooth solution $v$ to (NS) we show that if $\Delta v \in L^{\frac65} (\Bbb R^3)$ and $\lim_{|x|\to \infty} v(x)=0$, then $v=0$. For the extreme case of the self-similar Euler equations, which is equivalent to the steady Euler equations with damping, we prove that if there exists $q\in [\frac92, \infty)$ such that $v\in L^q(\Bbb R^3)$, then $v=0$. For the general self-similar Euler equations we present an improved version of Theorem 1.1 in \cite{cha0} with a simplified proof.

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