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

Abstract

This work establishes a global well-posedness result for the three dimensional axisymmetric Boussinesq system with critical rough initial data. Specifically, we aim at extending the results in our previous paper [22] to the case where the initial vorticity and density are finite Radon measures. The roadmap towards proving our claim is based upon the strategy in [22, 17]. Namely, performing a fixed point argument to build up a unique local solution, then, showing that the constructed solution is in fact global in time. Nevertheless, crucial arguments from [22], required for the construction of the local solution, need to be carefully analyzed when the datum, and more precisely when the initial density, is not a Lebesgue-integrable function. This is where several techniques of measure theory come into play. Accordingly, we first develop numerous notions on axisymmetric measures, in a general context. These techniques are employed thereafter to understand the coupling between some important quantities of the system and, eventually, to construct the desired solution whenever only the atomic parts of the initial measures are small enough.