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Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system

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  • In this paper we deal with the initial-boundary value problem for chemotaxis-fluid model involving more complicated nonlinear coupling term, precisely, the following self-consistent system

    $\left\{ \begin{array}{l}{n_t} + u \cdot \nabla n = \Delta {n^m} - \nabla \cdot (n\nabla c) + \nabla \cdot (n\nabla \phi ), \;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\{c_t} + u \cdot \nabla c = \Delta c - nc, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\{u_t} + (u \cdot \nabla )u + \nabla P = \Delta u - n\nabla \phi + n\nabla c, \;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \\\nabla \cdot u = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(x, t) \in \Omega \times (0, T), \end{array} \right.$

    where $Ω\subset \mathbb{R}^2$ is a bounded domain with smooth boundary.

    The novelty here is that both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid is considered, which leads to the stronger coupling than usual chemotaxis-fluid model studied in the most existing literatures. To the best of our knowledge, there is no global solvability result on this chemotaxis-Navier-Stokes system in the past works. It is proved in this paper that global weak solutions exist whenever $m>1$ and the initial data is suitably regular. This extends a result by Di Francesco, Lorz and Markowich (Discrete Cont. Dyn. Syst. A 28 (2010)) which asserts global existence of weak solutions under the constraint $m∈(\frac{3}{2}, 2]$ in the corresponding Stokes-type simplified system.

    Mathematics Subject Classification: 35K55, 35Q92, 35Q35, 92C17.


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