Chemotaxis-Fluid coupled Model for swimming Bacteria with Nonlinear Diffusion: Global Existence and Asymptotic Behavior

We study the system ct+u⋅∇c=Δc−nf(c) nt+u⋅∇n=Δnm−∇⋅(nχ(c)∇c) ut+u⋅∇u+∇P−ηΔu+n∇ϕ=0 ∇⋅u=0. 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∈(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∈( m*,2] with m*>3/2, due to the use of classical Sobolev inequalities.