Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

By Michael Winkler. Calculus of Variations and Partial Differential Equations, 2015

291 citations recorded by the source corpus.

We consider the chemotaxis-fluid system 0.1in a bounded convex 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, where ϕ∈W1,∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \in W^{1,\infty }(\Omega )$$\end{document} and D, f and S are given functions with values in [0,∞),[0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty ), [0,\infty )$$\end{document} and R3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{3\times 3}$$\end{document}, respectively. In the existing literature, the derivation of results on global existence and qualitative behavior essentially relies on the use of energy-type functionals which seem to be available only in special situations, necessarily requiring the matrix-valued S to actually reduce to a scalar function of c which, along with f, in addition should satisfy certain quite restrictive structural conditions. The present work presents a novel a priori estimation method which allows for removing any such additional hypothesis: besides appropriate smoothness assumptions, in this paper it is only required that f is locally bounded in [0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )$$\end{document}, that S is bounded in Ω×[0,∞)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \times [0,\infty )^2$$\end{document}, and that D(n)≥kDnm-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D(n)\ge k_{D}n^{m-1}$$\end{document} for all n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} with some kD>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{D}>0$$\end{document} and some m>76.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} m>\frac{7}{6}. \end{aligned}$$\end{document} It is shown that then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a globally defined weak solution. The method introduced here is efficient enough to moreover provide global boundedness of all solutions thereby obtained in that, inter alia, n∈L∞(Ω×(0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in L^\infty (\Omega \times (0,\infty ))$$\end{document}. Building on this boundedness property, it can finally even be proved that in the large time limit, any such solution approaches the spatially homogeneous equilibrium (n0¯,0,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\overline{n_0},0,0)$$\end{document} in an appropriate sense, where n0¯:=1|Ω|∫Ωn0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{n_0}:=\frac{1}{|\Omega |} \int _{\Omega }n_0$$\end{document}, provided that merely n0≢0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_0\not \equiv 0$$\end{document} and f>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f>0$$\end{document} on (0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,\infty )$$\end{document}. To the best of our knowledge, these are the first results on boundedness and asymptotics of large-data solutions in a three-dimensional chemotaxis-fluid system of type (0.1).

Subfields: Cell Biology, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/fa7fc4752fbcff607bff949a7e5a51fb1f978e0a

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Citations in the corpus

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”
Long-term behaviour in a chemotaxis-fluid system with logistic source (2016, background): “Other variants of the model that are commonly treated include nonlinear (porous medium type) diffusion of bacteria, where ∆n is replaced by ∆n 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 co”
Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux (2017, background): “Then, with the help of Theorem 2.2, we can straightforwardly deduce the uniform Hölder properties of c , ∇c and u by the standard parabolic regularity theory as the proof of Lemma 3.18 and Lemma 3.19 in Winkler [41].”
The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system (2017, unknown): “Contrary to this, for the corresponding chemotaxis-Stokes variant with D(n) = nm−1, n > 0, it could be shown that global weak solutions, yet known to exist whenever m ≥ 1 [8,37], are locally bounded in ̄ × [0,∞) if m > (8)7 [30] and even globally bounded if m > (7)6 [39].”
Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement (2017, methodology): “4) as sufficient for global existence and boundedness of weak solutions to an associated no-flux-no-fluxDirichlet initial-boundary value problem for all reasonably regular initial data in three-dimensional bounded convex domains ([46], cf.”
Stabilization in a two-dimensional chemotaxis-Navier–Stokes system (2013, unknown): “…appropriate assumptions on χ and f [10], but beyond this one can secondly prove local-in-time and even global boundedness of these solutions in the cases m > 87 and m 76 , respectively, and thereby rule out the occurrence of blow-up in ﬁnite time, and also in inﬁnite time (cf. [ 33] and [42] .”
Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary (2021, result): “Following [28], we fix ðvdÞd2ð0, 1Þ C1 0 ðXÞ such that 0 vd 1 in X and vd % 1 in X as d & 0, and rewriting (2.”
Global classical small-data solutions for a three-dimensional chemotaxis Navier–Stokes system involving matrix-valued sensitivities (2016, background): “An alternative assumption prompting the existence of weak solutions in the 3D-Stokes-setting is that of nonlinear diffusion of bacteria, that is, with ∆n replaced by ∇ · (n∇n), with an exponent m > 76 , [61].”