Rigidity of steady solutions to the Navier–Stokes equations in high dimensions and its applications

Abstract

The study of solutions with scale-invariant bounds plays an important role in understanding the regularity theory for the Navier–Stokes equations. In this paper, we investigate steady-state Navier–Stokes solutions in higher dimensions and prove that any steady solution in ${\mathbb{R}}^n$ satisfying $|\mathbf{u}(x)|\le C/|x|$ in ${\mathbb{R}}^n\setminus \{0\}$ must be trivial when $n\ge 4$. Neither smallness assumptions on $C$ nor self-similarity assumptions on $\mathbf{u}$ are required. The result can be used to show that a singularity at the origin with $|\mathbf{u}(x)|=O(1/|x|)$ for $x$ near 0 is removable in all dimensions $n\ge 4$. It can also be used (for $n\ge 4$) to identify the explicit leading-order terms and optimal asymptotic behavior of solutions exhibiting the critical decay rate $O(1/|x|)$. Our main idea in proving the rigidity result is to use weighted energy estimates and a sign property of the total head pressure, which make it possible to break the scaling.