Chemotaxis fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior

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 that the adiabatic exponent is an element of (3/2,2], such system features global in time solutions in two space dimensions for large data. Moreover, in the case the adiabatic exponent equals 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 linear diffusion case. The quadratic case 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 adiabatic exponents, due to the use of classical Sobolev inequalities.