Global Well-Posedness for the 2D Keller-Segel-Navier-Stokes System with Fractional Diffusion

In this paper, we consider Cauchy problem for the 2D incompressible Keller-Segel-Navier-Stokes equations with the fractional diffusion $\{\begin{array}{rl}& {\partial }_{t}n+u\cdot \mathrm{\nabla }n-\mathrm{\Delta }n=-\mathrm{\nabla }\cdot (n\mathrm{\nabla }c)-n^3,\\ & {\partial }_{t}c+u\cdot \mathrm{\nabla }c-\mathrm{\Delta }c=-c+n,\\ & {\partial }_{t}u+u\cdot \mathrm{\nabla }u+{\wedge }^{2\alpha }u+\mathrm{\nabla }P=-n\mathrm{\nabla }\varphi ,\end{array}$ \(\begin{aligned} \left \{ \begin{aligned} &\partial _{t}n+u\cdot \nabla n-\Delta n=-\nabla \cdot (n\nabla c)- n^{3}, \\ &\partial _{t}c+u\cdot \nabla c-\Delta c=-c+n, \\ &\partial _{t}u+u\cdot \nabla u+\wedge ^{2\alpha }u+\nabla P=-n\nabla \phi , \end{aligned} \right . \end{aligned}\) where $\wedge :={(-\mathrm{\Delta })}^{\frac{1}{2}}$ $\wedge :=(-\Delta )^{\frac{1}{2}}$ and $\alpha \in [\frac{1}{2},1]$ $\alpha \in [\frac{1}{2},1]$ . We get the global well-posedness for the above system with the rough initial data by a new priori estimate of the solutions.