On some Liouville type theorems for the compressible Navier-Stokesequations

Dong Li; Xinwei Yu

doi:10.3934/dcds.2014.34.4719

Article Contents

2014, Volume 34, Issue 11: 4719-4733. Doi: 10.3934/dcds.2014.34.4719

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On some Liouville type theorems for the compressible Navier-Stokes equations

Dong Li^1, and
Xinwei Yu^2,

1.
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T1Z2
2.
Department of Mathematical and Statistical Sciences, University of Alberta, 632 CAB, Edmonton, AB T6G 2G1

Received: January 31, 2013

Revised: January 31, 2014

Published: October 2014

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Abstract

We prove several Liouville type results for stationary solutions of the $d$-dimensional compressible Navier-Stokes equations. In particular, we show that when the dimension $d ≥ 4$, the natural requirements $\rho \in L^{\infty} ( \mathbb{R}^d )$, $v \in \dot{H}^1 (\mathbb{R}^d)$ suffice to guarantee that the solution is trivial. For dimensions $d=2,3$, we assume the extra condition $v \in L^{\frac{3d}{d-1}}(\mathbb R^d)$. This improves a recent result of Chae [1].

Keywords:

Compressible Navier-Stokes,
Liouville.

Mathematics Subject Classification: 35Q35.

Citation:

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References

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