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Singular sensitivity in a Keller–Segel-fluid system

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Abstract

In bounded smooth domains \(\Omega \subset \mathbb {R}^N\), \(N\in \{2,3\}\), considering the chemotaxis–fluid system

$$\begin{aligned} n_t + u\cdot \nabla n&= \Delta n - \chi \nabla \cdot \left( \frac{n}{c}\nabla c\right) \\ c_t + u\cdot \nabla c&= \Delta c - c + n\\ u_t + \kappa (u\cdot \nabla ) u&= \Delta u + \nabla P + n\nabla \phi \end{aligned}$$

with singular sensitivity, we prove global existence of classical solutions for given \(\phi \in C^2(\overline{\Omega })\), for \(\kappa =0\) (Stokes-fluid) if \(N=3\) and \(\kappa \in \{0,1\}\) (Stokes- or Navier–Stokes-fluid) if \(N=2\) and under the condition that

$$\begin{aligned} 0<\chi <\sqrt{\frac{2}{N}}. \end{aligned}$$

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Authors and Affiliations

  1. Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098, Paderborn, Germany

    Tobias Black & Johannes Lankeit

  2. Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

    Masaaki Mizukami

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  1. Tobias Black
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Correspondence to Johannes Lankeit.

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Black, T., Lankeit, J. & Mizukami, M. Singular sensitivity in a Keller–Segel-fluid system. J. Evol. Equ. 18, 561–581 (2018). https://doi.org/10.1007/s00028-017-0411-5

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