  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.
