Besov Mixed-Morrey Spaces: On an Application to the Navier–Stokes Equations

Abstract In this paper we introduce two new classes of functional spaces, namely, Besov mixed-Morrey spaces and Fourier–Besov mixed-Morrey spaces, and then we establish some basic properties for these classes. Moreover, we explore the d -dimensional incompressible Navier–Stokes equations in this context, by a Bony’s paraproduct approach, in order to get the global well-posedness for small initial data. These results provide a new class of initial data with a sort of anisotropy in relation to its spatial variables or frequency variables.