  We consider the stationary (time-independent) Navier-Stokes equations in the
whole threedimensional 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 decaying 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. Contents
