# Navier-Stokes equations with external forces in Besov-Morrey spaces

## TL;DR

This paper proves the existence and uniqueness of local solutions to the Navier-Stokes equations with external forces in Besov-Morrey spaces, extending to global solutions under small initial data and forces, using adapted methods from Besov space analysis.

## Contribution

It introduces the analysis of Navier-Stokes equations within homogeneous Besov-Morrey spaces, extending previous methods from Besov spaces to this broader setting.

## Key findings

- Existence and uniqueness of local strong solutions
- Global solutions possible with small initial data and external forces
- Extension of Besov space methods to Besov-Morrey spaces

## Abstract

We establish the existence and uniqueness of local strong solutions to the Navier-Stokes equations with arbitrary initial data and external forces in the homogeneous Besov-Morrey space. The local solutions can be extended globally in time provided the initial data and external forces are small. We adapt the method introduced in \cite{ks6}, where the Besov space is considered, to the setting of the homogeneous Besov-Morrey space.