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A two-dimensional Keller–Segel–Navier–Stokes system with logarithmic sensitivity: generalized solutions and classical solutions

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Abstract

This paper is concerned with the following Keller–Segel-fluid model

$$\left\{ \begin{array}{ll} n_t+u\cdot \nabla n=\Delta n-\chi \nabla \cdot (\frac{n}{c}\nabla c),&{}x\in \Omega ,~t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+n,&{}x\in \Omega ,~t>0,\\ u_t+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \Phi ,&{}x\in \Omega ,~t>0, \end{array}\right. (\star )$$

in a smoothly bounded planar domain. It is shown that under no-flux/no-flux/Dirichlet boundary conditions and for appropriately regular initial data, an associated initial-boundary problem possesses a globally defined solution in a suitably generalized sense whenever \(\chi >0.\) Beyond this, for the same initial-boundary problem, a uniquely classical solution can be established for any \(\chi \in (0,\chi _{*})\) for some \(\chi _{*}>1,\) which extends a previous result derived from a completely alternative approach under a stronger hypothesis.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Acknowledgements

The author would like to express his thanks to the anonymous referee for his helpful comments. This work is supported by the National Natural Science Foundation of China (Grant No. 11901298), and is partially supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20221508).

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11901298), and is partially supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20221508).

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  1. College of Sciences, Nanjing Agricultural University, Nanjing, 210095, People’s Republic of China

    Ji Liu

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Communicated by Laszlo Szekelyhidi.

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Liu, J. A two-dimensional Keller–Segel–Navier–Stokes system with logarithmic sensitivity: generalized solutions and classical solutions. Calc. Var. 62, 23 (2023). https://doi.org/10.1007/s00526-022-02371-9

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