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

Abstract

We study Liouville type of theorems for the Navier-Stokes and the Euler equations on ${\mathbb{R}}^N$, $N\ge 2$. 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 ${\int }_{{\mathbb{R}}^N}p(x,t)w_2(x)dx\ge 0$ 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 ${\mathbb{R}}^N$, $N\ge 3$.

17 pages

Cited by in corpus (1)

• On the nonexistence of stationary weak solutions to the compressible fluid equations