Daniel F. Machado | Searcharxiv

arXiv2023

On the well-posedness in Besov-Herz spaces for the inhomogeneous incompressible Euler equations

In this paper we study the inhomogeneous incompressible Euler equations in the whole space $\mathbb{R}^n$ with $n\geq3$. We obtain well-posedness and blow-up results in a new framework for inhomogeneous fluids, more precisely Besov-Herz spaces that are Besov spaces based on Herz ones, covering particularly critical cases of the regularity. Comparing with previous works on Besov spaces, our results provide a larger initial data class for a well-defined flow. For that, we need to obtain suitable linear estimates for some conservation-law models in our setting such as transport equations and the linearized inhomogeneous Euler system.

math.AP↗

arXiv2022

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 $n\geq 3$. 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.

math.AP↗