CHEMOTAXIS-FLUID COUPLED MODEL FOR SWIMMING BACTERIA WITH NONLINEAR DIFFUSION: GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR

We study the system {c(t) + u center dot del c = Delta c - nf(c) n(t) + u center dot del n = Delta n(m) - del center dot (n chi(c)del c) u(t) + u center dot del u + del P - eta Delta u + n del phi = 0 del center dot 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 is an element of (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 is an element of (m*, 2} with m* > 3/2, due to the use of classical Sobolev inequalities.