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

TL;DR

This paper introduces new Besov mixed-Morrey and Fourier-Besov mixed-Morrey spaces, establishing their properties and applying them to prove global well-posedness of Navier-Stokes equations with anisotropic initial data.

Contribution

The paper develops novel functional spaces and applies them to Navier-Stokes equations, demonstrating global well-posedness for small anisotropic initial data.

Findings

Introduction of Besov mixed-Morrey and Fourier-Besov mixed-Morrey spaces

Establishment of basic properties of these spaces

Proof of global well-posedness for Navier-Stokes with anisotropic initial data

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 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.