Global well-posedness for the 2D Boussinesq equations with partial temperature-dependent dissipative terms

In this paper, we obtain the global existence and uniqueness results for the 2D Boussinesq equations with mixed partial temperature-dependent viscosity and thermal diffusivity. Because of the deficiency of the dissipation in some directions, there is not enough smooth effect in those directions, which makes the problem of regularity become more difficult. In order to overcome these difficulties, we construct some crucial inequalities and the global estimates for the higher fractional derivatives of temperature in the necessary directions.