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Global weak solutions to a chemotaxis-Navier-Stokes system in $ \mathbb{R}^3 $

  • *Corresponding author: Kyungkeun Kang

    *Corresponding author: Kyungkeun Kang 

K. Kang is supported by NRF Grant No. 2019R1A2C1084685. J. Lee is supported by NRF Grant No. 2021R1A2C1092830. M. Winkler acknowledges support of the Deutsche Forschungsgemeinschaft (Project No. 462888149)

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  • The Cauchy problem in $ \mathbb{R}^3 $ for the chemotaxis-Navier–Stokes system

    $ \begin{eqnarray*} \left\{ \begin{array}{l} n_t + u\cdot\nabla n = \Delta n - \nabla \cdot (n\nabla c), \\ c_t + u\cdot\nabla c = \Delta c - nc, \\ u_t + (u\cdot\nabla) u = \Delta u + \nabla P + n\nabla\phi, \qquad \nabla \cdot u = 0, \ \end{array} \right. \end{eqnarray*} $

    is considered. Under suitable conditions on the initial data $ (n_0, c_0, u_0) $, with regard to the crucial first component requiring that $ n_0\in L^1( \mathbb{R}^3) $ be nonnegative and such that $ (n_0+1)\ln (n_0+1) \in L^1( \mathbb{R}^3) $, a globally defined weak solution with $ (n, c, u)|_{t = 0} = (n_0, c_0, u_0) $ is constructed. Apart from that, assuming that moreover $ \int_{ \mathbb{R}^3} n_0(x) \ln (1+|x|^2) dx $ is finite, it is shown that a weak solution exists which enjoys further regularity features and preserves mass in an appropriate sense.

    Mathematics Subject Classification: 92C17, 35Q30, 35D30, 35K55.


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