Navier-Stokes Equations with Fractional Dissipation in Sum of Pseudomeasure Spaces

In this paper is studied the local well-posedness of the Navier-Stokes system with initial data belonging to a sum of two pseudomeasure-type spaces denoted by $PM^{a,b}:=PM^{a}+PM^{b}.$ New results about local well-posedness and regularity of mild solutions of Navier-Stokes system are obtained. The proof requires to show an interesting H\"{o}lder-type inequality in $PM^{a,b}$, as well as to establish estimates of the semigroup generated by fractional power of \ Laplacian $(-\Delta)^{\gamma}$ on these spaces.