Besov mixed-Morrey spaces: On an Application to the Navier-Stokes Equations

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 mean of a Bony's paraproduct approach, in order to get the global well-posedness for small initial data. These results provides a new class of initial data with a sort of anisotropy in relation to its spatial variables or frequency variables.