On the global well-posedness of a class of Boussinesq-Navier-Stokes systems

In this paper we consider the following 2D Boussinesq-Navier-Stokes systems partial derivative(t)u+u.del u+del p=-nu|D|(alpha)u+theta e(2) partial derivative(t)theta+u.del theta=-kappa|D|(beta)theta divu=0 with nu > 0, kappa > 0 and 0 < beta < alpha < 1. When 6-root 6/1(=0.888) < alpha < 1, 1 - alpha < beta <= f(alpha), where f(alpha) < 1 is an explicit function as a technical bound, we prove the global well-posedness results for the rough initial data.