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 $n \geq 3$. 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.