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Global solutions to the coupled chemotaxis-fluids system in a 3D unbounded domain with boundary

Yingping Peng

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P. R. China

E-mail Address: yingping_peng@163.com

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Zhaoyin Xiang

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P. R. China

E-mail Address: zxiang@uestc.edu.cn

Corresponding author

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https://doi.org/10.1142/S0218202518500239Cited by:32 (Source: Crossref)

Abstract

In this paper, we investigate the global existence of solutions to a coupled chemotaxis-fluids system in a three-dimensional unbounded domain with boundary. In the chemotaxis-Navier–Stokes case, we establish the global existence and uniqueness of strong solutions around a constant state, while in the chemotaxis-Stokes case, we show the global existence of weak solution for large initial cell density and velocity. Our proof is based on some uniform a priori estimates obtained by using the anisotropic $L^{p}$ technique and the elliptic estimates. Trading time derivative and spatial derivative is one of our highlights too. To the best of our knowledge, this is the first analytical work for the well-posedness of chemotaxis-fluids system in an unbounded domain with boundary, which is a first step toward a qualitative theory for the free boundary problem of chemotaxis-fluids system. Our results are consistent with the experiment observation and numerical simulation.

Communicated by M. Winkler

Keywords:

AMSC: 35K55, 35Q92, 35Q35, 92C17

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