Differential and Integral Equations

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

Dongho Chae

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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 ${\mathbb R}^N$, $N\geq3$.

Article information

Source
Differential Integral Equations, Volume 25, Number 3/4 (2012), 403-416.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356012741

Mathematical Reviews number (MathSciNet)
MR2917889

Zentralblatt MATH identifier
1265.35245

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 35Q35: PDEs in connection with fluid mechanics 76Dxx: Incompressible viscous fluids 76Bxx: Incompressible inviscid fluids

Citation

Chae, Dongho. On the Liouville type theorems with weights for the Navier-Stokes equations and Euler equations. Differential Integral Equations 25 (2012), no. 3/4, 403--416. https://projecteuclid.org/euclid.die/1356012741


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