Properties of the linear non-local Stokes operator and its application

In this paper, we first investigate some basic properties of the linear non-local Stokes operator, such as the structure of the Oseen tensor , maximal regularity and so on. By the special structure of the spherically symmetric -stable Lévy processes, we can decompose as where is smooth for with decaying as . Combining this decomposition with the Fourier multipliers theorem, we establish the maximal regularity theory of the non-local Stokes operator. Secondly, as an important application to the decomposition (0.1), we derive the precise asymptotic behavior at infinity for the small self-similar solution of the Navier–Stokes equations with a fractional diffusion. (paper)