Liouville type theorems for the fractional Navier-Stokes equations without the integrability condition of velocity in $\mathbb{R}^3$

TL;DR

This paper establishes Liouville type theorems for fractional Navier-Stokes equations in three dimensions, showing solutions must be trivial under certain boundedness and convergence conditions without requiring velocity integrability.

Contribution

It proves that smooth solutions converging to a nonzero constant at infinity are trivial, without assuming velocity integrability, using advanced harmonic analysis techniques.

Findings

Solutions must be trivial if they converge to a nonzero constant at infinity.

No integrability condition on velocity is necessary for the Liouville theorem.

Utilizes Lizorkin's multiplier theorem and commutator estimates for the fractional Navier-Stokes system.

Abstract

Motivated by the classification of solutions of harmonic functions, we investigate Liouville type theorems for the fractional Navier-Stokes equations in $\mathbb{R}^3$ under some conditions on the boundedness of fractional derivatives. We prove that the smooth solution must be a trivial solution provided that it uniformly converges to a nonzero constant vector at infinity by applying Lizorkin's multiplier theorem to establish \(L^p\) estimates for the fractional linear Oseen system and Coifman-McIntosh-Meyer type commutator estimates for the dissipation term. It is noteworthy that the integrability of velocity is not required here.