Global well-posedness of incompressible Bénard problem with zero dissipation or zero thermal diffusivity

In this paper, we establish the global well-posedness of the classical solution for the two-dimensional Bénard problem with zero dissipation or zero thermal diffusivity. Our work is partially motivated by the results about Boussinesq equations with zero dissipation or zero thermal diffusivity in Chae (2006) and Hou and Li (2005).