On the global existence for the axisymmetric Euler–Boussinesq system 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 v⁰∈B^5/2_2,1(R³) and ρ⁰∈B^1/2_2,1(R³)∩L^p(R³) 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 (see [2]) is not known to be valid and to circumvent this difficulty we use in a crucial way some geometric properties of the vorticity.