Global existence and uniqueness for a non linear Boussinesq system in   dimension two
  We study the global well-posedness of a two-dimensional Boussinesq system which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion of type $|\DD|^{\alpha}$ for the temperature. We prove that for $\alpha>1,$ there exists a unique global solution for initial data with critical regularities. 
Samira Sulaiman
math.AP