On some Liouville Type Theorems for the Compressible Navier-Stokes Equations

TL;DR

This paper establishes Liouville type theorems for stationary solutions of the compressible Navier-Stokes equations, showing under certain conditions that solutions must be trivial, with improvements over previous results.

Contribution

It extends Liouville theorems to higher dimensions and relaxes conditions needed for triviality of solutions, improving upon prior work by Chae (2012).

Findings

Solutions are trivial in dimensions d ≥ 4 under bounded density and finite energy.

Additional integrability conditions are required for dimensions 2 and 3.

The results generalize and strengthen previous Liouville theorems for compressible Navier-Stokes equations.

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 $d \geqslant 4$, the natural requirements $\rho \in L^{\infty} (\mathbbm{R}^d)$, $v \in \dot{H}^1 (\mathbbm{R}^d)$ 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 (2012).