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

$ c_t+u \cdot \nabla c = \Delta c- nf(c) $
$ n_t + u \cdot \nabla n = \Delta n^m- \nabla \cdot (n \chi(c)\nabla c) $
$ u_t + u \cdot \nabla u + \nabla P - \eta\Delta u + n \nabla \phi=0 $
$\nabla \cdot 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 $m\in(3/2, 2]$ 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 $m\in$( m^*$,2]$ with m^*>3/2, due to the use of classical Sobolev inequalities.