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

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

2008-11-28

arxiv.org/abs/0811.4647v2

Mathematics

Analysis of PDEs

17 pages

Scientific paper

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

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