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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
- Ferrari, A: On the blow-up of solutions of 3-D Euler equations in a bounded domain. Commun. Math. Phys. 155, 277-294 (1993) MathSciNetView ArticleMATHGoogle Scholar
- Keller, E, Segel, L: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399-415 (1970) View ArticleMATHGoogle Scholar
- Keller, E, Segel, L: Model for chemotaxis. J. Theor. Biol. 30, 225-234 (1971) View ArticleMATHGoogle Scholar
- Keller, E, Segel, L: Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 235-248 (1971) View ArticleMATHGoogle Scholar
- Biler, P: Global solutions to some parabolic-elliptic systems of chemotaxis. Adv. Math. Sci. Appl. 9, 347-359 (2009) MathSciNetMATHGoogle Scholar
- Corrias, L, Perthame, B, Zaag, H: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1-28 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Hillen, T, Painter, K: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183-217 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Horstmann, D: From 1970 until present: the Keller-Segel model in chemotaxis and its consequences: I. Jahresber. Dtsch. Math.-Ver. 105, 103-165 (2003) MathSciNetMATHGoogle Scholar
- Sleeman, B, Ward, M, Wei, J: Existence, stability, and dynamics of spike patterns in a chemotaxis model. SIAM J. Appl. Math. 65, 790-817 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Winkler, M: Global solutions in a fully parabolic chemotaxis system with singular sensitivity. Math. Methods Appl. Sci. 34, 176-190 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Wrzosek, D: Long time behaviour of solutions to a chemotaxis model with volume filling effect. Proc. R. Soc. Edinb., Sect. A, Math. 136, 431-444 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Chae, M, Kang, K, Lee, J: Global existence and temporal decay in Keller-Segel models coupled to fluid equations. Commun. Partial Differ. Equ. 39(7), 1205-1235 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Fan, J, Zhao, K: Global dynamics of a coupled chemotaxis-fluid model on bounded domains. J. Math. Fluid Mech. 16(2), 351-364 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Xie, H, Ma, C: On blow-up criteria for a coupled chemotaxis fluid model. J. Inequal. Appl. 2017, 30 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Amann, H: Maximal regularity for nonautonomous evolution equations. Adv. Nonlinear Stud. 4, 417-430 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Bourguignon, J, Brezis, H: Remarks on the Euler equation. J. Funct. Anal. 15, 341-363 (1974) MathSciNetView ArticleMATHGoogle Scholar
- Shirota, T, Yanagisawa, T: A continuation principle for the 3D Euler equations for incompressible fluids in a bounded domain. Proc. Jpn. Acad., Ser. A, Math. Sci. 69, 77-82 (1993) View ArticleMATHGoogle Scholar