## Differential and Integral Equations

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

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

#### 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