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

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 (−∆) α 2 with 0 < α < 2. By applying a fixed-point argument, weak solutions can be obtained in the Sobolev space H˙α2 (R3) and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of α that the zero function is the unique smooth solution. The additional integrability condition is almost sharp for 3/5 < α < 5/3. Moreover, in the case 1 < α < 2 a gain of regularity is established under some conditions, although the study of regularity in the regime 0 < α ≤ 1 seems for the moment to be an open problem.