Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions and its applications

Abstract

Solutions with scaling-invariant bounds such as self-similar solutions, play an important role in the understanding of the regularity and asymptotic structures of solutions to the Navier-Stokes equations. In this paper, we prove that any steady solution satisfying $|\Bu(x)|\leq C/|x|$ for any constant $C$ in $\mathbb{R}^n\setminus \{0\}$ with $ n \geq 4$, must be zero. Our main idea is to analyze the velocity field and the total head pressure via weighted energy estimates with suitable multipliers so that the proof is pretty elementary and short. These results not only give the Liouville-type theorem for steady solutions in higher dimensions with neither smallness nor self-similarity type assumptions, but also help to remove a class of singularities of solutions and give the optimal asymptotic behaviors of solutions at infinity in the exterior domains.