A coupled anisotropic chemotaxis-fluid model: The case of two-sidedly degenerate diffusion

In this article, the mathematical analysis of a model arising from biology consisting of diffusion, chemotaxis with volume filling effect and transport through an incompressible fluid, is studied. Motivated by numerical and modeling issues, the global-in-time existence of weak solutions to this model is investigated. The novelty with respect to other related papers lies in the presence of two-sidedly nonlinear degenerate diffusion and of anisotropic and heterogeneous diffusion tensors where we prove global existence for Chemotaxis-Navier-Stokes system in space dimensions less or equal than four and we show uniqueness of weak solutions for Chemotaxis-Stokes system in two or three space dimensions under further assumptions.