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

Abstract

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.