Saturation of the signal on the boundary: Global weak solvability in a chemotaxis-Stokes system with porous-media type cell diffusion

In this paper, we investigate a chemotaxis-Stokes system with porous-media type cell diffusion ∇⋅(nm−1∇n) in a bounded domain Ω⊂RN (N=2,3) with smooth boundary. We prove that for the no-flux–saturation–no-slip boundary value and suitable regular initial data, the mild assumption m>3N−22N is sufficient for the global existence and boundedness of weak solutions, and in particular confirm the experimental observation in Tuval et al. (2005, PNAS). In comparison to the considerable literature, the novelty here is that we require signal to attain a prescribed saturation value throughout the boundary. This result will be the first step towards a qualitative theory for the boundary layer.