On the Liouville type theorems with weights for the Navier-Stokes equations and Euler equations

Abstract

We deduce Liouville type theorems for the Navier-Stokes and the Euler equations on ${\mathbb R}^N$, $N\geq 2$. Specifically, we prove that if a weak solution $(v,p)$ satisfies $|v|^2 +|p| \in L^1 (0,T; L^1({\mathbb R}^N, w_1(x)dx))$ and $\int_{{\mathbb R}^N} p(x,t)w_2 (x)dx \geq0$ for some weight functions $w_1(x)$ and $w_2 (x)$, then the solution is trivial, namely $v=0$ almost everywhere on ${\mathbb R}^N \times (0, T)$. Similar results hold for the MHD equations on ${\mathbb R}^N$, $N\geq3$.