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The fast signal diffusion limit in Keller–Segel(-fluid) systems

  • Yulan Wang
  • Michael Winkler
  • Zhaoyin XiangEmail author
Article

Abstract

This paper deals with convergence of solutions to a class of parabolic Keller–Segel systems, possibly coupled to the (Navier–)Stokes equations in the framework of the full model
$$\begin{aligned} \left\{ \begin{array}{lll} \partial _t n_\varepsilon + u_\varepsilon \cdot \nabla n_\varepsilon &{}=&{} \Delta n_\varepsilon - \nabla \cdot \Big ( n_\varepsilon S(x,n_\varepsilon ,c_\varepsilon )\cdot \nabla c_\varepsilon \Big ) + f(x,n_\varepsilon ,c_\varepsilon ), \\ \varepsilon \partial _t c_\varepsilon + u_\varepsilon \cdot \nabla c_\varepsilon &{}=&{} \Delta c_\varepsilon - c_\varepsilon + n_\varepsilon , \\ \partial _t u_\varepsilon + \kappa (u_\varepsilon \cdot \nabla )u_\varepsilon &{}=&{} \Delta u_\varepsilon + \nabla P_\varepsilon + n_\varepsilon \nabla \phi , \qquad \nabla \cdot u_\varepsilon =0 \end{array}\right. \end{aligned}$$
to solutions of the parabolic–elliptic counterpart formally obtained on taking \(\varepsilon {\searrow } 0\). In smoothly bounded physical domains \(\Omega \subset {{\mathbb {R}}}^{N}\) with \(N\ge 1\), and under appropriate assumptions on the model ingredients, we shall first derive a general result which asserts certain strong and pointwise convergence properties whenever asserting that supposedly present bounds on \(\nabla c_\varepsilon \) and \(u_\varepsilon \) are bounded in \(L^\lambda ((0,T);L^q(\Omega ))\) and in \(L^\infty ((0,T);L^r(\Omega ))\), respectively, for some \(\lambda \in (2,\infty ]\), \(q>N\) and \(r>\max \{2,N\}\) such that \(\frac{1}{\lambda }+\frac{N}{2q}<\frac{1}{2}\). To our best knowledge, this seems to be the first rigorous mathematical result on a fast signal diffusion limit in a chemotaxis-fluid system. This general result will thereafter be concretized in the context of two examples: firstly, for an unforced Keller–Segel–Navier–Stokes system we shall establish a statement on global classical solutions under suitable smallness conditions on the initial data, and show that these solutions approach a global classical solution to the respective parabolic–elliptic simplification. We shall secondly derive a corresponding convergence property for arbitrary solutions to fluid-free Keller–Segel systems with logistic source terms, which in spatially one-dimensional settings turn out to allow for a priori estimates compatible with our general theory. Building on the latter in conjunction with a known result on emergence of large densities in the associated parabolic–elliptic limit system, we will finally discover some quasi-blowup phenomenon for the fully parabolic Keller–Segel system with logistic source and suitably small parameter \(\varepsilon >0\).

Mathematics Subject Classification

92C17 (primary) 35Q30 35K55 35B65 35Q92 (secondary) 

Notes

Acknowledgements

The authors are very grateful to the referee for his/her detailed comments and valuable suggestions, which greatly improved the manuscript. Y. Wang was supported by the Applied Fundamental Research Plan of Sichuan Province (No. 2018JY0503) and Xihua Scholars Program of Xihua University. M. Winkler acknowledges support of the Deutsche Forschungsgemeinschaft (Project No. 411007140, GZ: WI 3707/5-1). Z. Xiang was partially supported by the NNSF of China under Grants 11571063 and 11771045, and by the Fundamental Research Funds for the Central Universities (No. ZYGX2019J096).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of ScienceXihua UniversityChengduChina
  2. 2.Institut für MathematikUniversität PaderbornPaderbornGermany
  3. 3.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduChina

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