Global boundedness and eventual regularity of chemotaxis-fluid model driven by porous medium diffusion

The chemotaxis-fluid model was proposed by Goldstein et al. in 2005 to characterize the bacterial swimming phenomenon in incompressible fluid. For the three-dimensional case, the global existence of bounded solutions to chemotaxis(–Navier)–Stokes model has always been an open problem. Therefore, researchers have been led to seek alternative avenues by turning their attention to the model with slow diffusion ($\Delta n^m$ with $m \gt 1$). Even with slow diffusion, the problem is not easy to solve. In particular, the closer $m$ is to $1$, the more difficult the study becomes. In this paper, we put forward a new method to prove the global existence and boundedness of weak solutions for any $m \gt 1$. The new method allows us to obtain higher regularity when $m$ is close to 1. Subsequently, we also prove that the weak solution converges to the constant equilibrium point $(\overline n_0, 0, 0)$ in the sense of $L^\infty$-norm for $1\lt m \leq \frac{5}{3}$. Based on this, we prove that the weak solution becomes smooth after a certain time and eventually becomes a classical solution.