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

TL;DR

This paper establishes local and global well-posedness results for the inhomogeneous Navier-Stokes equations in Besov-Morrey spaces, allowing for initial data with discontinuities and expanding the class of initial conditions for which solutions exist.

Contribution

It introduces a new analytical framework using Besov-Morrey spaces for inhomogeneous fluids, enabling larger initial data classes and handling discontinuous densities.

Findings

Global-in-time flow constructed under smallness conditions

Allows initial densities with discontinuities

Provides estimates for heat semigroup localization and transport equations

Abstract

In this paper we consider the incompressible inhomogeneous Navier-Stokes equations in the whole space with dimension $n\geq 3$. 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 $\mathcal{N}_{p,q,\infty }^{n/p}\cap L^{\infty }$ 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.