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

2008/11/28 by Dongho Chae, Chae, Dongho
Mathematics · Engineering · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.0811.4647

Abstract

We study Liouville type of theorems for the Navier-Stokes and the Euler equations on \Bbb R^N, N≥ 2. Specifically, we prove that if a weak solution (v,p) satisfies |v|² +|p| ∈ L¹ (0,T; L¹(\Bbb R^N, w₁(x)dx)) and ∫_{\Bbb R^N} p(x,t)w₂ (x)dx ≥0 for some weight functions w₁(x) and w₂ (x), then the solution is trivial, namely v=0 almost everywhere on \Bbb R^N × (0, T). Similar results hold for the MHD Equations on \Bbb R^N, N≥3.

Citations

• Some mathematical questions related to the mhd equations

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