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A regularity criterion for the Keller-Segel-Euler system
© The Author(s) 2017
Received: 12 May 2017
Accepted: 24 June 2017
Published: 29 August 2017
We consider a Keller-Segel-Euler model and prove a regularity criterion of the local strong solutions in a 3D bounded domain Ω.
On the other hand, when \(u=0\), system (1.3) and (1.4) reduces to the classical Keller-Segel chemotaxis model [2–4], which received many studies [5–11] on well-posedness and pattern formation of solutions.
2 Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. Since local existence results can be proved by using standard arguments, say, Galerkin method, we only deal with the a priori estimates.
This completes the proof of Theorem 1.1.
We consider the 3D Keller-Segel-Euler system in a bounded domain. It is a challenging open problem whether the local solution exists globally. Here, a regularity criterion in terms of the vorticity and oxygen concentration is established to guarantee smoothness up to time T. It will help people to gain understanding of the model. We hope to find more inside structures and establish refined regularity criteria.
The first, second and fourth authors are partially supported by the National Natural Science Foundation of China (Grant No. 11171154 and 11501346). The third author extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).
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