Global existence of weak solutions for a Keller-Segel-fluid model with nonlinear diffusion

We consider the Cauchy problem for a Keller-Segel-fluid model with degenerate diffusion for cell density, which is mathematically formulated as a porus medium type of Keller-Segel equations coupled to viscous incompressible fluid equations. We establish the global-in- time existence of weak solutions and bounded weak solutions depending on some conditions of parameters such as chemotactic sensitivity and consumption rate of oxygen for certain range of diffusive exponents of cell density in two and three dimensions.