Notes on Liouville-type theorems for the 3D stationary Navier-Stokes equations

TL;DR

This paper extends Liouville-type theorems for the 3D stationary Navier-Stokes equations within variable Lebesgue spaces, identifying new regions where these theorems remain valid under specific conditions.

Contribution

It generalizes previous results by establishing Liouville-type theorems in additional non-negligible regions for variable exponent spaces.

Findings

Identified two new regions where Liouville theorems hold

Extended the validity of these theorems beyond previous exponent ranges

Generalized prior results in variable Lebesgue space framework

Abstract

In \cite{CV23}, Chamorro and Vergara-Hermosilla established several Liouville-type theorems to the Navier-Stokes equations in the framework of the variable Lebesgue spaces. These results may allow the variable exponent $p(\cdot)$ beyond the range of $[3,\frac{9}{2}]$ in some non-negligible regions in $\mathbb{R}^3$. In this paper we find two new non-negligible regions, in which the Liouville-type theorems still hold under some assumptions imposed on $p(\cdot)$ in these regions. Our results can be regarded as the generalization of the results in \cite{CV23}.