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 (\Bbb R^N))$ with $\int_{\Bbb R^N} p(x,t)dx \geq 0$, then the corresponding velocity should be trivial, namely $v=0$ on $\Bbb R^N \times (0,T)$. In particular, this is the case when $p\in L^1 (0,T; \mathcal{H}^1 (\Bbb R^N))$, where $\mathcal{H}^1 (\Bbb R^N)$ the Hardy space. On the other hand, we have equipartition of energy over each component, if $p\in L^1 (0,T; L^1 (\Bbb R^N))$ with $\int_{\Bbb R^N} p(x,t)dx <0$. Similar results hold also for the magnetohydrodynamic equations.