Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities

By Xinru Cao, Johannes Lankeit. Calculus of Variations and Partial Differential Equations, 2016

210 citations recorded by the source corpus.

The coupled chemotaxis fluid system nt=Δn-∇·(nS(x,n,c)·∇c)-u·∇n,(x,t)∈Ω×(0,T),ct=Δc-nc-u·∇c,(x,t)∈Ω×(0,T),ut=Δu-(u·∇)u+∇P+n∇Φ,∇·u=0,(x,t)∈Ω×(0,T),∇c·ν=(∇n-nS(x,n,c)·∇c)·ν=0,u=0,(x,t)∈∂Ω×(0,T),n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),x∈Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lll} n_t=\Delta n-\nabla \cdot (n S(x,n,c)\cdot \nabla c)-u\cdot \nabla n, &{}\quad (x,t)\in \Omega \times (0,T),\\ c_t=\Delta c-nc-u\cdot \nabla c, &{}\quad (x,t)\in \Omega \times (0,T),\\ u_t=\Delta u-(u\cdot \nabla )u+\nabla P+n\nabla \Phi ,\quad \nabla \cdot u=0, &{}\quad (x,t)\in \Omega \times (0,T),\\ \nabla c\cdot \nu =(\nabla n-nS(x,n,c)\cdot \nabla c)\cdot \nu =0, \;\; u=0,&{}\quad (x,t)\in \partial \Omega \times (0,T),\\ n(x,0)=n_{0}(x),\quad c(x,0)=c_{0}(x),\quad u(x,0)=u_0(x), &{}\quad x\in \Omega , \end{array} \right. \end{aligned}$$\end{document}where S∈(C2(Ω¯×[0,∞)2))N×N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\in (C^2({\overline{\Omega }}\times [0,\infty )^2))^{N\times N}$$\end{document}, is considered in a bounded domain Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^N$$\end{document}, N∈{2,3}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in \{2,3\}$$\end{document}, with smooth boundary. We show that it has global classical solutions if the initial data satisfy certain smallness conditions and give decay properties of these solutions.

Subfields: Cell Biology, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/71b8af5f00083fda4b1d5337a82f892b4ce68690

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Citations in the corpus

Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity (2018, background): “[34] and [4] we first verify a respective weak solution property by showing that − ∫ ∞”
A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization (2019, background): “For such models, numerous results have been achieved in the past few years, addressing topics ranging from the mere existence of global solutions ([13], [73], [3], [6]), also in the context of nonlinear cell diffusion ([14], [55], [60], [9], [77]), to global boundedness ([66], [5], [24], [4]) and ev”
Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary (2021, background): “Extensions in various directions are concerned with global classical solvability for small data [13, 14], or address modified systems involving nonlinear cell diffusion of porous medium type [15–19], or also accounting for saturation effects in chemotactic migration [20–22].”
The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system (2017, unknown): “In full chemotaxis-Navier–Stokes systems, further boundedness properties could only be established upon imposing appropriate smallness conditions on the initial data [2,4,5,7,18,26,42].”
Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components (2018, background): “also [4], [2], [34], [35] and [43] for extensions involving fluid interaction and addressing three-dimensional settings).”
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption (2016, background): “1] or [2] for a recent contribu tion with an extensive bibliography).”
Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement (2017, background): “This may be viewed as part of a more comprehensive ambition to understand the destabilizing potential of chemotaxis-fluid interaction that has been in the focus also of studies including further relevant processes such as logistic proliferation and death, or off-diagonal cross-diffusive migration ([”
Long-term behaviour in a chemotaxis-fluid system with logistic source (2016, background): “replaced by ∆nm for some m>1 (see [34, 35, 10, 7, 49]), thereby improving chances for ﬁnding bounded solutions, or, exchanging χ∇ · (n∇c) for ∇· (nS(n,c,x)∇c), more complex sensitivity functions S([46, 40, 39, 16, 5]), which may be matrix-valued, thus introducing new mathematical challenges by de”
Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D (2017, background): “e also been shown to exist in [35], whereas many results treating variants of (1.2) in three-dimensional frameworks are again restricted to weak solutions emanating from small initial data (e.g. [13],[4]). Nevertheless, even in theses cases, where global regularity is hard to prove, some results con”