An \(L^p\)-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 \((-\Delta )^{\alpha /2}\) ${(-\mathrm{\Delta })}^{\alpha /2}$ 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 \(\alpha \) $\alpha$ 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 \(\alpha \) $\alpha$ . 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 \(\alpha =2\) $\alpha =2$ ) stationary Navier–Stokes equations.