Global existence for an attraction-repulsion chemotaxis fluid model with logistic source

Abelardo Duarte-Rodríguez; Lucas C. F. Ferreira; Élder J. Villamizar-Roa

doi:10.3934/dcdsb.2018180

Article Contents

2019, Volume 24, Issue 2: 423-447. Doi: 10.3934/dcdsb.2018180

This issue Previous Article Existence and uniqueness of solutions of free boundary problems in heterogeneous environments Next Article Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains

Global existence for an attraction-repulsion chemotaxis fluid model with logistic source

1.
Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia
2.
Universidade Estadual de Campinas, Departamento de Matemática-IMECC, CEP 13083-859, Campinas-SP, Brazil

* Corresponding author: Élder J. Villamizar-Roa
* Corresponding author: Élder J. Villamizar-Roa

Received: October 31, 2017

Revised: December 31, 2017

Early access: May 2018

Published: January 2019

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

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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.

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Mathematics Subject Classification: Primary: 35Q92, 92C17; Secondary: 35Q35, 35K55, 76D05.

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