## Abstract

We study the system c_{t} + u · ∇_{c} = ∇_{c} -nf(c) n_{t} + u · ∇_{n} = ∇_{n}^{m} - ∇ · (n×(c) ∇_{c}) u_{t} + u·∇u + ∇P - η∇_{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 ε (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 ε (m^{*}, 2] with m^{*} > 3/2, due to the use of classical Sobolev inequalities.

Original language | English (US) |
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Pages (from-to) | 1437-1453 |

Number of pages | 17 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 28 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2010 |

Externally published | Yes |

## Keywords

- Chemotaxis model
- Nonlinear diffusion
- Stokes equations

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics