Rigidity of Stationary Navier-Stokes Solutions with Transverse bound in Dimensions Five and Higher

Abstract

For $n\geq5$, let $(u,p)$ be a smooth solution of the stationary incompressible Navier-Stokes equations in $\R^n\setminus\{x'=0\}$, where $\{x'=0\}$ is the $x_n$-axis. We prove that the scale-invariant transverse bound $|u(x)|\leq C |x'|^{-1}$ forces $u\equiv0$. This result relies on a quantitative local regularity theorem, which shows that the transverse bound condition yields uniform control of $u$ and $\nabla u$ in a smaller ball and removes the possible singularity along the axis. The proof combines weak extension across the axis, approximate Green functions for an adjoint drift operator, a one-sided bound for the total head pressure, a localized Frehse-R\r{u}\v{z}i\v{c}ka weighted estimate, and a finite Stokes-Morrey bootstrap. Applying the local theorem at arbitrarily large scales yields whole-space rigidity and removability of line singularities. This together with the asymptotic expansions on the exterior domains in the case $|u(x)|\leq C |x|^{-1}$ in [1], yields the same expansion in exterior domains under the condition $|u(x)|\leq C |x'|^{-1}$.