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

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.