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February 2019, 24(2): 423-447. doi: 10.3934/dcdsb.2018180
Global existence for an attraction-repulsion chemotaxis fluid model with logistic source
Abelardo Duarte-Rodríguez 1, , Lucas C. F. Ferreira 2, and Élder J. Villamizar-Roa 1,,
Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia
Universidade Estadual de Campinas, Departamento de Matemática-IMECC, CEP 13083-859, Campinas-SP, Brazil
* Corresponding author: Élder J. Villamizar-Roa
Received November 2017 Revised January 2018 Published June 2018
Fund Project: The second author has been partially supported by CNPq and FAPESP, Brazil. The third author has been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, and Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, contrato Colciencias FP 44842-157-2016. The authors would like to thank an anonymous referee for useful remarks and suggestions.
We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of $\mathbb{R}^N,$ $N = 2, 3,$ for small initial data in $L^p$-spaces.
Keywords: Chemotaxis, Keller-Segel-Navier-Stokes system, attraction-repulsion, logistic source, global mild solutions.
Mathematics Subject Classification: Primary: 35Q92, 92C17; Secondary: 35Q35, 35K55, 76D05.
Citation: Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
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Abelardo Duarte-Rodríguez Lucas C. F. Ferreira Élder J. Villamizar-Roa