Liouville type of theorems for the Euler and the Navier-Stokes equations

Abstract

We prove Liouville type of theorems for weak solutions of the Navier-Stokes and the Euler equations. In particular, if the pressure satisfies $p\in L^1(0,T;L^1({\mathbb{R}}^N))$ with ${\int }_{{\mathbb{R}}^N}p(x,t)dx\ge 0$, then the corresponding velocity should be trivial, namely $v=0$ on ${\mathbb{R}}^N\times (0,T)$. In particular, this is the case when $p\in L^1(0,T;{\mathcal{H}}^1({\mathbb{R}}^N))$, where ${\mathcal{H}}^1({\mathbb{R}}^N)$ the Hardy space. On the other hand, we have equipartition of energy over each component, if $p\in L^1(0,T;L^1({\mathbb{R}}^N))$ with ${\int }_{{\mathbb{R}}^N}p(x,t)dx<0$. Similar results hold also for the magnetohydrodynamic equations.

15 pages