Dynamics of Partial Differential Equations

Volume 19 (2022)

Number 4

Global existence and asymptotic behavior of solutions for a fractional chemotaxis-Navier-Stokes system

Pages: 285 – 309

DOI: https://dx.doi.org/10.4310/DPDE.2022.v19.n4.a3

Authors

Miguel A. Fontecha-Medina (Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia)

Élder J. Villamizar-Roa (Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia)

Abstract

We consider a fractional chemotaxis-Navier-Stokes model in the whole space $\mathbb{R}^N , N \geq 2$, with a time-fractional variation in the Caputo sense, a fractional self-diffusion for the physical variables and a fractional dissipation mechanism for the chemoattraction process. We prove the existence and uniqueness of global mild solutions with small initial data in a larger class of critical spaces of Besov–Morrey type. Our result extend the well-posedness ones in the classical (no fractional regime) obtained by Postigo and Ferreira [16]. We also prove the long-time asymptotic stability of solutions.

Keywords

chemotaxis-Navier-Stokes system, Besov–Morrey spaces, Caputo fractional derivative, fractional dissipation

2010 Mathematics Subject Classification

35A01, 35B40, 35K55, 35Q92, 35R11, 92C17

Full Text (PDF format)

M.A. Fontecha-Medina was supported by Vicerrectoría de Investigación y Extensión.

E. J. Villamizar-Roa was supported by Vicerrectoría de Investigación y Extensión, UIS, Project 3704.

Received 26 January 2022

Published 14 December 2022