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

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November 2014, 34(11): 4719-4733. doi: 10.3934/dcds.2014.34.4719

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, Canada
2.	Department of Mathematical and Statistical Sciences, University of Alberta, 632 CAB, Edmonton, AB T6G 2G1

Received February 2013 Revised February 2014 Published May 2014

Abstract
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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: 35Q3.

Citation: Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719

References:

[1]	D. Chae, Remarks on the liouville type results for the compressible navier-stokes equations in $\mathbb{R}^N2$, Nonlinearity, 25 (2012), 1345-1349. doi: 10.1088/0951-7715/25/5/1345.
[2]	E. Feireisl, Dynamics of Viscous Compressible Fluids, volume 26 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2004.
[3]	J. Jost, Partial Differential Equations, Graduate Texts in Mathematics, 214. Springer, New York, 2007. doi: 10.1007/978-0-387-49319-0.
[4]	P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10. Oxford University Press, New York, 1998.
[5]	A. Novotny and I. Stra, Introduction to the Mathematical Theory of Compressible Flow, volume 27 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2004.

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References:

[1]	D. Chae, Remarks on the liouville type results for the compressible navier-stokes equations in $\mathbb{R}^N2$, Nonlinearity, 25 (2012), 1345-1349. doi: 10.1088/0951-7715/25/5/1345.
[2]	E. Feireisl, Dynamics of Viscous Compressible Fluids, volume 26 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2004.
[3]	J. Jost, Partial Differential Equations, Graduate Texts in Mathematics, 214. Springer, New York, 2007. doi: 10.1007/978-0-387-49319-0.
[4]	P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10. Oxford University Press, New York, 1998.
[5]	A. Novotny and I. Stra, Introduction to the Mathematical Theory of Compressible Flow, volume 27 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2004.

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