On weak solutions of nonstationary boussinesq equations

We study weak solutions of the initial and boundary value problems of the Boussinesq equations which describe the natural convection in a viscous incompressible fluid. We construct a global weak solution for the initial velocity in L² and the initial temperature in L¹. We show that the temperature θ(x, t) of our weak solution is Hölder continuous in x for almost every t > 0. In general, it is not known whether weak solutions are unique or not. We show that weak solutions are unique if they are in some Lebesgue space. We show, moreover, that weak solutions are regular if they belong to the uniqueness class.