American Institute of Mathematical Sciences

December  2010, 28(4): 1437-1453. doi: 10.3934/dcds.2010.28.1437

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

 1 Department of Pure and Applied Mathematics, Via Vetoio, Loc. Coppito, 67100 L’Aquila, Italy 2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom, United Kingdom

Received  October 2009 Revised  February 2010 Published  June 2010

We study the system

$c_t+u \cdot \nabla c = \Delta c- nf(c)$
$n_t + u \cdot \nabla n = \Delta n^m- \nabla \cdot (n \chi(c)\nabla c)$
$u_t + u \cdot \nabla u + \nabla P - \eta\Delta u + n \nabla \phi=0$
$\nabla \cdot u = 0.$

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.

Citation: Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437
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