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- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer Nature B.V. 2021
- ISSN
- 0167-8019
- eISSN
- 1572-9036
- DOI
- 10.1007/s10440-021-00436-z
- Publisher site
- See Article on Publisher Site
Abstract
This paper studies a coupled chemotaxis-Navier-Stokes model with arbitrary superlinear dampening logistic term {nt+u⋅∇n=Δn−∇⋅(n∇c)+rn−μnα,ct+u⋅∇c=Δc−nc,ut+(u⋅∇)u=Δu+∇P+n∇Φ,∇⋅u=0(⋆)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document} $$\begin{aligned} \left \{ \textstyle\begin{array}{l} n_{t}+u\cdot \nabla n=\Delta n-\nabla \cdot (n\nabla c)+ rn-\mu n^{ \alpha }, \\ c_{t}+u\cdot \nabla c=\Delta c-nc, \\ u_{t}+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \Phi , \quad \nabla \cdot u=0 \end{array}\displaystyle \right .\quad \quad (\star ) \end{aligned}$$ \end{document} in a bounded domain Ω⊂R3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\Omega \subset \mathbb{R}^{3}$\end{document} with smooth boundary, under homogeneous Neumann boundary conditions for n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n$\end{document} and c\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$c$\end{document}, and homogeneous Dirichlet boundary condition for u\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u$\end{document}. Here Φ∈C1+β(Ω‾)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\Phi \in C^{1+\beta }(\overline{\Omega })$\end{document}, r≥0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$r \ge 0$\end{document}, μ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mu >0$\end{document} and α>1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha >1$\end{document}. It is shown in Wang (Math. Models Methods Appl. Sci. 30(06):1217–1252, 2020) that (⋆) possesses at least one global weak solution for 1<α<2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$1<\alpha < 2$\end{document} and the solution becomes sufficiently smooth after finite time if we require 65≤α<2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\frac{6}{5}\le \alpha < 2$\end{document}. The present work proves the eventual smoothness property holds for 1<α<2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$1<\alpha <2$\end{document}.
Journal
Acta Applicandae Mathematicae – Springer Journals
Published: Oct 1, 2021
Keywords: Chemotaxis; Navier-Stokes; Global existence; Eventual smoothness