New Liouville type theorems for the stationary Navier-Stokes equations
Wenke Tan
TL;DR
The authors address Liouville-type questions for the homogeneous stationary Navier–Stokes equations in $\mathbb{R}^3$, including the fractional case, under finite Dirichlet energy. They derive a new Dirichlet integral identity that expresses $\int_{\mathbb{R}^3}|\nabla u|^2dx$ (and its fractional analogue $\int_{\mathbb{R}^3}|(-\Delta)^{s/2}u|^2dx$) purely from low-frequency information at the origin in frequency space via Littlewood–Paley/Bony decompositions. Using this localization, they obtain Liouville-type theorems without the far-field decay assumption, under bounds on low-frequency components such as $\liminf_{k\to-\infty}2^{-k}\|\dot{S}_k u\|_{L^{\infty}}<\infty$ (and analogous Besov/ Fourier bounds), with sharp results at $s=\tfrac{5}{6}$ and $s=\tfrac12$. The work also provides high-frequency localization results for $\tfrac12<s<\tfrac{5}{6}$ and fully treats the case $s=\tfrac{5}{6}$, including a separate treatment of $s=\tfrac12$, thereby unifying the classical and fractional stationary Navier–Stokes theories through a frequency-space Liouville framework.
Abstract
We mainly research the Liouville type problem for the stationary Navier-Stokes equations (including the fractional case) in $\mathbb{R}^3$. We first establish a new formula for the Dirichlet integral of solutions and show that the globally defined quantity $\int_{\mathbb{R}^3}|\nabla u|^2dx$ is completely determined by the information of the solution $u$ at the origin in frequency space. From this character, we show some new Liouville type theorems for solutions of the stationary Navier-Stokes equations. Then we extend the obtained results for classical stationary Navier-Stokes equations to the stationary fractional Navier-Stokes equations for $\frac{1}{2}\leq s<1$, especially, we solve the Liouville type problem for $s=\frac{5}{6}$.