On the global well-posedness of a generalized 2D Boussinesq equations

In this paper, we consider the global solutions to a generalized 2D Boussinesq equation \begin{align*} \left \{\begin{aligned} & \partial_{t} \omega + u\cdot \nabla \omega + \nu \Lambda^{\alpha} \omega = \theta_{x_{1}} , \quad \\ & u = \nabla^{\bot} \psi = (-\partial_{x_{2}} , \partial_{x_{1}}) \psi , \quad \Delta \psi = \Lambda^{\sigma} (\log (I-\Delta))^{\gamma} \omega , \quad \\ & \partial_{t} \theta + u\cdot \nabla \theta + \kappa \Lambda^{\beta} \theta = 0, \quad \\ & \omega(x,0) = \omega_{0}(x) , \quad \theta(x,0) = \theta_{0}(x), \end{aligned}\right. \end{align*} with $\sigma \geq 0$, $\gamma \geq 0$, $\nu >0$, $\kappa>0$, $\alpha < 1$ and $\beta < 1$. When $\sigma = 0$, $\gamma \geq 0$, $\alpha \in [0.95,1)$ and $\beta \in (1-\alpha,g(\alpha))$, where $g(\alpha)<1$ is an explicit function as a technical bound, we prove that the above equation has a global and unique solution in suitable functional space.