On the fractional chemotaxis Navier-Stokes system in the critical spaces
type: Item
instance of: scholarly article
title: On the fractional chemotaxis Navier-Stokes system in the critical spaces
zbMATH Open document ID: 1505.35347
author: Joelma Azevedo
author: Claudio Cuevas
author: Jarbas Dantas
author: Clessius Silva
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2022-10-10T00:00:00Z
review text: The authors deal with the double chemotaxis model of fractional order under the effect of the Navier-Stokes fluid \begin{align*} &^cD^\alpha_tn+u\cdot\nabla n=\Delta n-\nabla\cdot(n\nabla c)-\nabla\cdot(n\nabla v),&&\quad \text{in } \mathbb{R}^N\times(0,\infty), \\ &^cD^\alpha_tc+u\cdot\nabla c=\Delta c-nc,&&\quad \text{in } \mathbb{R}^N\times(0,\infty), \\ &^cD^\alpha_tv+u\cdot\nabla v=\Delta v-\gamma v+n,&&\quad \text{in } \mathbb{R}^N\times(0,\infty), \\ &^cD^\alpha_tu+(u\cdot\nabla) u=\Delta u-\nabla p-nf,&&\quad\text{in } \mathbb{R}^N\times(0,\infty), \\ &\nabla\cdot u=0,&&\quad \text{in } \mathbb{R}^N\times(0,\infty), \\ &n(x,0)=n_0(x),c(x,0)=c_0(x),v(x,0)=v_0(x),u(x,0)=u_0(x),&&\quad \text{in } \mathbb{R}^N, \end{align*} where \(N\ge2\), \(\gamma\ge0\), \(^cD^\alpha_t\) is the Caputo fractional derivative of order \(\alpha\in(0, 1)\). They prove the existence of global mild solutions with the small critical initial data in Besov-Morrey spaces. The results can obtain the self-similar solutions provided the initial data are homogeneous functions with small norms and considering the case of chemical attractant without degradation rate. Moreover, they show the asymptotic stability of solutions as the time goes to infinity and obtain a class of asymptotically self- similar ones.
reviewed by: Yuanyuan Ke
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 92C17
zbMATH DE Number: 7599027
zbMATH Keywords: chemotaxis models
zbMATH Keywords: incompressible Navier-Stokes equations
zbMATH Keywords: well-posedness
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: critical Besov-Morrey spaces
MaRDI profile type: Publication
full work available at URL: dcdsb.2022088
OpenAlex ID: W4285131399
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DOI: 10.3934/DCDSB.2022088
description: scientific article; zbMATH DE number 7599027