Bounded weak and strong time periodic solutions to a three-dimensional chemotaxis-Stokes model with porous medium diffusion

In this paper, we study the time periodic problem to a three-dimensional chemotaxis-Stokes model with porous medium diffusion $Delta n^m$ and inhomogeneous mixed boundary conditions. By using a double-level approximation method and some iterative techniques, we obtain the existence and time-space uniform boundedness of weak time periodic solutions for any $m>1$. Moreover, we improve the regularity for $mlefrac{4}{3}$ and show that the obtained periodic solutions are in fact strong periodic solutions.