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

Wendong Wang; Guoxu Yang; Jianbo Yu

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arXiv · 2505.04895

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

Wendong Wang·Guoxu Yang·Jianbo Yu

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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.

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Wendong Wang, Guoxu Yang, Jianbo Yu. 2025-05-08. Liouville type theorems for the fractional Navier-Stokes equations without the integrability condition of velocity in $\mathbb{R}^3$. https://arxiv.org/abs/2505.04895

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