On the Global Existence for the Axisymmetric Euler-Boussinesq System in Critical Besov Spaces

TL;DR

This paper establishes global existence and uniqueness for the 3D axisymmetric Boussinesq system with initial data in critical Besov spaces, utilizing geometric vorticity properties to overcome the lack of Beale-Kato-Majda criterion.

Contribution

It proves global well-posedness for the axisymmetric Boussinesq system in critical Besov spaces, a significant extension beyond previous results.

Findings

Global existence and uniqueness for the system.

Use of geometric vorticity properties.

Handling of initial data in critical Besov spaces.

Abstract

This paper is devoted to the global existence and uniqueness results for the three-dimensional Boussinesq system with axisymmetric initial data $v^{0}{\in}B_{2,1}^{5/2}(\RR^3)$ and$ ${\rho}^{0}{\in}B_{2,1}^{1/2}(\RR^3)\cap L^{p}(\RR^3)$ with $p>6.$ This system couples the incompressible Euler equations with a transport-diffusion equation governing the density. In this case the Beale-Kato-Majda criterion is not known to be valid and to circumvent this difficulty we use in a crucial way some geometric properties of the vorticity.