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

TL;DR

This paper proves Liouville-type theorems for stationary Navier-Stokes equations in three dimensions, showing trivial solutions under localized radial integrability conditions on the velocity.

Contribution

It introduces new Liouville results based on integrability of the radial velocity component, relaxing global conditions typically required.

Findings

Any $ ext{H}^1$ solution is trivial if the radial velocity component is in $L^p$ for $3/2 < p extless 3$.

Uniqueness is established with localized $L^6$-type conditions, approaching the critical exponent at infinity.

Radial and localized integrability conditions suffice to ensure rigidity of solutions.

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 $3/2 < p \leq 3$. 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.