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An optimal result for global existence in a three-dimensional Keller–Segel–Navier–Stokes system involving tensor-valued sensitivity with saturation

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Abstract

This paper focuses on the following Keller–Segel–Navier–Stokes system with rotational flux:

in a bounded domain \(\Omega \subset {\mathbb {R}}^3\) with a smooth boundary, where \(\kappa \in {\mathbb {R}}\) is a given constant, \(\phi \in W^{1,\infty }(\Omega )\), \(|S(x,n,c)|\le C_S(1+n)^{-\alpha }\), and the parameter \(\alpha \ge 0\). If \(\alpha >\frac{1}{3}\), then, for all reasonable regular initial data, a corresponding initial-boundary value problem for (KSNF) possesses a globally defined weak solution. This result improves upon the result of Wang (Math Models Methods Appl Sci 27(14):2745–2780, 2017), in which the global very weak solution for the system (KSNF) is obtained. In comparison with the result of the corresponding fluid-free system, the optimal condition on the parameter \(\alpha \) for global (weak) existence is established. Our proofs rely on a variant of the natural gradient-like energy functional.

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Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (No. 11601215) and the Shandong Provincial Science Foundation for Outstanding Youth (No. ZR2018JL005).

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Authors and Affiliations

  1. School of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of China

    Yuanyuan Ke

  2. School of Mathematics and Statistics Science, Ludong University, Yantai, 264025, People’s Republic of China

    Jiashan Zheng

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  1. Yuanyuan Ke
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Correspondence to Jiashan Zheng.

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Communicated by L. Caffarelli.

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Ke, Y., Zheng, J. An optimal result for global existence in a three-dimensional Keller–Segel–Navier–Stokes system involving tensor-valued sensitivity with saturation. Calc. Var. 58, 109 (2019). https://doi.org/10.1007/s00526-019-1568-2

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