On an almost sharp Liouville type theorem for fractional Navier-Stokes equations

Abstract

We investigate existence, Liouville type theorems and regularity results for the 3D stationary and incompressible fractional Navier-Stokes equations: in this setting the usual Laplacian is replaced by its fractional power $(-\Delta)^{\frac{\alpha}{2}}$ with $0<\alpha<2$. By applying a fixed point argument, weak solutions can be obtained in the Sobolev space $\dot{H}^{\frac{\alpha}{2}}(\mathbb{R})$ and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of $\alpha$ that the zero function is the unique smooth solution. The additional integrability condition is almost sharp for $3/5<\alpha<5/3$. Moreover, in the case $1<\alpha<2$ a gain of regularity is established under some conditions, however the study of regularity in the regime $0<\alpha\leq 1$ seems for the moment to be an open problem.