On the axisymmetric Boussinesq system with rough initial data

In this paper, we establish the global existence to the three-dimensional axisymmetric Boussinesq system with rough initial data, which is described as follows: (i) the initial vorticity or the initial density is a Dirac measure supported on a circle; (ii) both of them are concentrated within a circular region. Specifically, by virtue of the regularization procedure imposed on the initial data, we first construct a sequence of approximate solutions to the system. Subsequently, we derive the uniform estimates of the approximate solutions and employ compactness arguments to obtain the existence of a solution. Furthermore, we demonstrate that the solution converges weakly to the initial data in an appropriate sense.