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

TL;DR

This paper establishes the existence and regularity of weak and strong time periodic solutions for a 3D chemotaxis-Stokes model with porous medium diffusion, using approximation and iterative methods.

Contribution

It introduces a novel approach to prove the existence and regularity of periodic solutions in a complex chemotaxis-Stokes system with porous medium diffusion.

Findings

Existence of weak time periodic solutions for all m>1.

Improved regularity results for m ≤ 4/3.

Periodic solutions are shown to be strong solutions under certain conditions.

Abstract

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 $m\le\frac{4}{3}$ and show that the obtained periodic solutions are in fact strong periodic solutions.