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

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.