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

The full text of this article is unavailable through your IP address: 3.15.223.129

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