An Lp-theory for fractional stationary Navier-Stokes equations

We consider the stationary (time-independent) Navier-Stokes equations in the whole three-dimensional space, under the action of a source term and with the fractional Laplacian operator (-Delta)(alpha/2) in the diffusion term. In the framework of Lebesgue and Lorentz spaces, we find some natural sufficient conditions on the external force and on the parameter alpha to prove the existence and in some cases nonexistence of solutions. Secondly, we obtain sharp pointwise decay rates and asymptotic profiles of solutions, which strongly depend on alpha. Finally, we also prove the global regularity of solutions. As a bi-product, we obtain some uniqueness theorems so-called Liouville-type results. On the other hand, our regularity result yields a new regularity criterion for the classical (i.e. with alpha = 2) stationary Navier-Stokes equations.