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In this paper, we study the time periodic problem to a three-dimensionalchemotaxis-Stokes model with porous medium diffusion $\Delta n^m$ andinhomogeneous mixed boundary conditions. By using a double-level approximationmethod and some iterative techniques, we obtain the existence and time-spaceuniform boundedness of weak time periodic solutions for any $m>1$. Moreover, weimprove the regularity for $m\le\frac{4}{3}$ and show that the obtainedperiodic solutions are in fact strong periodic solutions.

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