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

TL;DR

This paper establishes well-posedness and blow-up results for the inhomogeneous incompressible Euler equations in Besov-Herz spaces, expanding the class of initial data for which solutions are well-defined, especially in critical regularity cases.

Contribution

It introduces a new analytical framework using Besov-Herz spaces for inhomogeneous fluids, extending previous results by covering larger initial data classes and critical regularities.

Findings

Proves well-posedness in Besov-Herz spaces for inhomogeneous Euler equations.

Establishes blow-up criteria within this new functional setting.

Provides linear estimates for transport and linearized Euler models in Besov-Herz spaces.

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.