Weak solutions for a bioconvection model related to Bacillus subtilis

We consider the initial-boundary value problem for the coupled Navier-Stokes-Keller-Segel-Fisher-Kolmogorov-Petrovskii-Piskunov system in two- and three-dimensional domains. The problem describes oxytaxis and growth of Bacillus subtilis in moving water. We prove existence of global weak solutions to the problem. We distinguish between two cases determined by the cell diffusion term and the space dimension, which are referred to as the supercritical and subcritical ones. In the first case, the choice of the kinetic function enjoys a wide range of possibilities: in particular, it can be zero. Our results are new even in the absence of the kinetic term. In the second case, the restrictions on the kinetic function are less relaxed: for instance, it cannot be zero but can be Fisher-like. In the case of linear cell diffusion, the solution is regular and unique provided the domain is the whole plane. In addition, we study the long-time behavior of the problem, find dissipative estimates, and construct attractors.

Keywords

bioconvection, oxytaxis, trajectory attractor, global solvability, Navier-Stokes, Keller-Segel, reaction-diffusion

2010 Mathematics Subject Classification

35B41, 35Q35, 35Q92, 76Zxx, 92C17

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Published 26 November 2013