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

We study the systemcₜ+u · ∇ c = Δ c- nf(c) n_t + u · ∇ n = Δ n^m- ∇ · (n χ(c)∇ c) uₜ + u · ∇ u + ∇ P - ηΔ u + 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.