On the Liouville type theorem for stationary compressible Navier-Stokes-Poisson equations in $\Bbb R^N$

TL;DR

This paper establishes Liouville type theorems for stationary solutions of the compressible Navier-Stokes-Poisson and Navier-Stokes equations in multi-dimensional space, showing under certain conditions that solutions must be trivial or constant.

Contribution

It proves new Liouville theorems for stationary solutions of NSP and NS equations under integrability and boundedness assumptions, extending known results to these systems.

Findings

Stationary solutions must be trivial (zero velocity) under conditions.

For Navier-Stokes, solutions with non-vacuum boundary conditions are constant.

Results hold in dimensions N ≥ 2.

Abstract

In this paper we prove Liouville type result for the stationary solutions to the compressible Navier-Stokes-Poisson equations(NSP) and the compressible Navier-Stokes equations(NS) in $\Bbb R^N$, $N\geq 2$. Assuming suitable integrability and the uniform boundedness conditions for the solutions we are led to the conclusion that $v=0$. In the case of (NS) we deduce that the similar integrability conditions imply $v=0$ and $\rho=$constant on $\Bbb R^N$. This shows that if we impose the the non-vacuum boundary condition at spatial infinity for (NS), $v\to 0$ and $\rho\to \rho_\infty >0$, then $v=0$, $\rho=\rho_\infty$ are the solutions.