Liouville-type theorem for the stationary inhomogeneous Navier-Stokes equations
Huiting Ding, Wenke Tan
TL;DR
This work proves a Liouville-type rigidity for stationary inhomogeneous Navier–Stokes flows in $\mathbb{R}^3$ by localizing the Dirichlet energy in frequency space using two stages. The approach decouples the key nonlinear interactions via paraproduct decompositions in the frequency domain, and shows the localized energy contributions vanish under $\rho\in L^{\infty}$, $\nabla u\in L^2$, and $\liminf_{k\to-\infty}\|\dot{S}_k u\|_{\dot{B}^{-1}_{\infty,\infty}}<\infty$, forcing $u=0$. This yields a uniqueness-type result determined by local frequency information, extending known Liouville theorems for homogeneous stationary NS to the inhomogeneous setting. The paper combines Littlewood-Paley theory, Bony's paraproducts, and energy estimates to relate local frequency bounds to global vanishing of the Dirichlet energy, with potential implications for frequency-domain regularity criteria in steady flows.
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 $ρu$ and $u$, the second one is to eliminate the non-zero frequency part coming from the interaction between $ρ$ and $u$. Based on the local formula of Dirichlet energy, we can establish suitable estimates on different frequency parts of $u$ and $ρ$, then show our new Liouville-type theorem.