Article Contents
Article Contents

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

• 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].
Mathematics Subject Classification: 35Q35.

 Citation:

•  [1] D. Chae, Remarks on the liouville type results for the compressible navier-stokes equations in $\mathbbR^N$, 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.