We study a system arising in the modelling of the motion of swimming bacteria under the effect of diffusion, oxygen-taxis and transport through an incompressible fluid. The novelty with respect to previous papers in the literature lies in the presence of nonlinear porous-medium-like diffusion in the equation for the density n of the bacteria, motivated by a finite size effect. We prove that, under the constraint m ε (3/2, 2] for the adiabatic exponent, such system features global in time solutions in two space dimensions for large data. Moreover, in the case m = 2 we prove that solutions converge to constant states in the large-time limit. The proofs rely on standard energy methods and on a basic entropy estimate which cannot be achieved in the case m = 1. The case m = 2 is very special as we can provide a Lyapounov functional. We generalize our results to the three-dimensional case and obtain a smaller range of exponents m ε (m, 2] with m > 3/2, due to the use of classical Sobolev inequalities.
Di Francesco, M., Lorz, A., & Markowich, P. (2010). Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete and Continuous Dynamical Systems, 28(4), 1437-1453. https://doi.org/10.3934/dcds.2010.28.1437