Liouville-type theorem for the stationary inhomogeneous Navier-Stokes equations

TL;DR

This paper proves a new Liouville-type theorem for the stationary inhomogeneous Navier-Stokes equations in three dimensions, using frequency localization techniques to analyze energy distribution and establish conditions for trivial solutions.

Contribution

It introduces a novel approach employing frequency localization to establish a Liouville-type theorem for inhomogeneous Navier-Stokes equations, extending previous results.

Findings

Established a Liouville-type theorem for stationary inhomogeneous Navier-Stokes

Developed a method to localize Dirichlet energy in frequency space

Provided estimates on frequency components of velocity and density

Abstract

In this manuscript, a new Liouville-type theorem for the three-dimensional stationary inhomogeneous Navier-Stokes equations is established. We first localize the Dirichlet energy into the region near the origin in frequency spaces by two times localizations. The first localization is to eliminate the non-zero frequency part coming from the interaction between $\rho u$ and $u$, the second one is to eliminate the non-zero frequency part coming from the interaction between $\rho$ and $u$. Based on the local formula of Dirichlet energy, we can establish suitable estimates on different frequency parts of $u$ and $\rho$, then show our new Liouville-type theorem.