Global-wellposedness of 2D Boussinesq equations with mixed partial temperature-dependent viscosity and thermal diffusivity

In this paper, we study the global well-posedness of 2D anisotropic nonlinear Boussinesq equations with horizontal temperature-dependent viscosity and vertical thermal diffusivity in the whole space. Due to lacking vertical viscosity and horizontal thermal diffusivity, there is no smooth effect in those directions. Besides, the nonlinearity of temperature-dependent viscosity gives rise to new difficulties. To solve it, we make full use of the incompressible condition and anisotropic inequalities to obtain the H1H1 estimates of velocity field and H1+sH1+s estimates of temperature for any s∈(0,1/2]s∈(0,1/2]. In the end, we build up a uniqueness criterion which together with the a priori estimates admits a unique global solution without any smallness assumptions.