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

Abstract

In this paper, we consider the global solutions to a generalized 2D Boussinesq equation $$\left \{ \begin{array}{ll}\partial_{t} \omega + u \cdot \nabla \omega + \nu \Lambda^{\alpha} \omega = \theta_{x_{1}} , \quad u = \nabla^{\bot} \psi = (-\partial_{x_{2}} , \partial_{x_{1}}) \psi , \quad \Delta \psi = \Lambda^{\sigma} (\log (I-\Delta))^{\gamma} \omega , \quad \partial_{t} \theta + u\cdot \nabla \theta + \kappa \Lambda^{\beta} \theta = 0, \quad \omega(x,0) = \omega_{0}(x) , \quad \theta(x,0) = \theta_{0}(x),\end{array}\right.$$ 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.