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

TL;DR

This paper establishes Liouville type theorems for weak solutions of the Navier-Stokes and Euler equations, showing conditions under which solutions must be trivial or exhibit energy equipartition, extending to magnetohydrodynamics.

Contribution

It introduces new Liouville theorems based on pressure integrability conditions, providing criteria for triviality or energy distribution in solutions.

Findings

Trivial solutions when pressure integral is non-negative.

Energy equipartition occurs when pressure integral is negative.

Results extend to magnetohydrodynamic 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.