Global well-posedness for the 2D Boussinesq equations with zero viscosity

We prove the global well-posedness of the two-dimensional Boussinesq equations with zero viscosity and positive diffusivity in bounded domains for rough initial data [u0∈L2, curlu0∈L∞ and θ0∈Bq,p2−2/p with p∈(1,∞), q∈(2,∞)]. Our method is based on the maximal regularity for heat equation.