Remarks on the global well-posedness of the axisymmetric Boussinesq system with rough initial data

TL;DR

This paper investigates the global well-posedness of the 3D axisymmetric viscous Boussinesq system with rough initial data, extending previous results to measure-type initial conditions and establishing conditions for global solutions.

Contribution

It introduces notions of axisymmetric measures and proves global well-posedness for initial measures with small atomic parts, advancing understanding of rough initial data in Boussinesq systems.

Findings

Global well-posedness for measure-type initial data with small atomic parts.

Extension of previous results to rough initial data of measure type.

Development of notions of axisymmetric measures in a general context.

Abstract

This work concerns the global well-posedness problem for the 3D axisymmetric viscous Boussinesq system with critical rough initial data. More precisely, we aim to extending our recent result \cite{Hanachi-Houamed-Zerguine} to the case of initial data of measure type. To this end, we should first develop some notions of axisymmetric measures in a general context, then, in the spirit of \cite{Gallay-Sverak}, we prove the global wellposedness result provided that the atomic parts of the initial measures are small enough.