Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component

Abstract

We study Liouville-type results for the stationary Navier--Stokes equations in ${\mathbb{R}}^3$. We prove that any ${\dot{H}}^1({\mathbb{R}}^3)$ solution is trivial under an integrability condition imposed only on the radial component of the velocity, namely $u_{\rho }(x)\in L^p({\mathbb{R}}^3)$ with $3/2<p\le 3$. We also establish a uniqueness result in a variable-exponent setting, where an $L^6$-type condition is required only on a bounded region, while the exponent approaches the critical value $3$ at infinity. Our analysis reveals that the rigidity of the stationary Navier--Stokes system can be driven by localized and radial integrability properties, rather than uniform global conditions.