Local and global analysis in Besov-Morrey spaces for inhomogeneous Navier-Stokes equations

In this paper we consider the incompressible inhomogeneous Navier-Stokes equations in the whole space with dimension $n\ge 3$. We present local and global well-posedness results in a new framework for inhomogeneous fluids, namely Besov-Morrey spaces ${\mathcal{N}}_{p,q,r}^s$ that are Besov spaces based on Morrey ones. In comparison with the previous works in Sobolev and Besov spaces, our results provide a larger initial-data class for both the velocity and density, constructing a unique global-in-time flow under smallness conditions on weaker initial-data norms. In particular, we can consider some kind of initial discontinuous densities, since our density class ${\mathcal{N}}_{p,q,\mathrm{\infty }}^{n\mathrm{/}p}\cap L^{\mathrm{\infty }}$ is not contained in any space of continuous functions. From a technical viewpoint, the Morrey underlying norms prevent the common use of energy-type and integration by parts arguments, and then we need to obtain some estimates for the localizations of the heat semigroup, the commutator, and the volume-preserving map in our setting, as well as estimates for transport equations and the linearized inhomogeneous Navier-Stokes system.