Global Boundedness to a 3D Chemotaxis–Stokes System with Porous Medium Cell Diffusion and General Sensitivity Under Dirichlet Signal Boundary Condition

Abstract

In this paper, we construct a globally bounded weak solution for the initial-boundary value problem of a three-dimensional chemotaxis–Stokes system with porous medium cell diffusion \(\Delta n^m\) and inhomogeneous Dirichlet signal boundary for each \(m>\frac{13}{12}\). Compared with the quite well-developed solvability for the no-flux signal boundary value with \(m>\frac{7}{6}\) (Winkler in Calc Var 54:3789–3828, 2015), to our best knowledge, this seems to be the first result on chemotaxis–fluid system with general matrix-valued sensitivity for such a Dirichlet signal boundary condition, under which even for the scalar sensitivity, we also extend the recent range \(m>\frac{7}{6}\) (Wu and Xiang in J Differ Equ 315:122–158, 2022). Our proof will be based on a new observation on the boundary estimate and on a three-step induction argument. The same technique can be applied to the two-dimensional setting to confirm a similar conclusion for any \(m>1\).