Navier–Stokes equations with external forces in Besov–Morrey spaces

In this paper, by adapting the method introduced in H. Kozono and S. Shimizu (Strong solutions of the Navier–Stokes equations based on the maximal Lorentz regularity theorem in Besov spaces. J Funct Anal. 2019;276:896–931), 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. Also, we show that the local solutions can be extended globally in time provided the initial data and external forces both are small.