Global regularity for the tropical climate model with fractional diffusion - Yang, Jing; Deng, Xuemei; Bie, Qunyi (AIMS mathematics, 2021)
In this paper, we investigate the following tropical climate model with fractional diffusion \begin{document}$ \begin{eqnarray} \left\{\begin{array}{ll} u_t+u\cdot\nabla u+\nabla p+\Lambda^{2\alpha}u+{\rm div}(v\otimes v) = 0,\\[1ex] v_t+u\cdot\nabla v+\nabla\theta+\Lambda^{2\beta}v+v\cdot\nabla u = 0,\\[1ex] \theta_t+u\cdot\nabla\theta+\Lambda^{2\gamma}\theta+{\rm div} v = 0,\\[1ex] {\rm div} u = 0,\\[1ex] ( u, v, \theta)(x,0) = ( u_0, v_0, \theta_0), \end{array} \right. \end{eqnarray} $\end{document} where $ (u_0, v_0, \theta_0) \in H^s(R^n) $ with $ s\geq 1, n\geq 3 $ and $ {\rm div} u_0 = 0 $. When the nonnegative constants $ \alpha, \beta $ and $ \gamma $ satisfy $ \alpha\geq\frac{1}{2}+\frac{n}{4}, \ \alpha+\beta\geq 1+\frac{n}{2}, \ \alpha+\gamma\geq1+\frac{n}{2} $, by using the energy methods, we obtain the global existence and uniqueness of solution for the system. In the special case $ \theta = 0 $, we could obtain the global solution provide that $ \alpha\geq\frac{1}{2}+\frac{n}{4}, \alpha+\beta\geq1+\frac{n}{2} $ and $ (u_0, v_0)\in H^s(s\geq1) $, which generalizes the existing result.