Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity
Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity
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DOI：
10.1007/s00526-015-0922-2
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发表时间：
2015-10
影响因子：
2.1
通讯作者：
M. Winkler
中科院分区：
文献类型：
--
作者：
M. Winkler
在边界光滑的有界凸区域上考虑趋化性流体系统0.1，其中D、f和Se分别是取值于和的给定函数。在现有文献中，关于整体存在性和定性行为的结果的推导基本上依赖于似乎只有在特殊情况下才可用的能量型泛函的使用，这必然要求矩阵值S实际上归结为标量函数，与f一起，还应满足某些相当严格的结构条件。本文提出了一种新的先验估计方法，它允许去掉任何这样的附加假设：除了适当的光滑性假设外，本文只要求f是局部有界的，S是有界的，和全部与一些和一些文件类[12pt]{minimal}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$\Begin{aliged}m>；\FRAC{7}{6}。证明了对于所有合理规则的初值，相应的初边值问题存在一个整体定义的弱解。这里介绍的方法是足够有效的，而且还提供了由此得到的所有解的全局有界性，特别是。在这种有界性的基础上，甚至可以证明，在大的时间范围内，任何这样的解在适当的意义上都逼近空间齐次平衡，其中，只要有足够的条件。据我们所知，这是关于三维趋化流体系统大数据解的有界性和渐近性的第一个结果。
We consider the chemotaxis-fluid system0.1in a bounded convex domainwith smooth boundary, whereandD,fandSare given functions with values inand, 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-valuedSto actually reduce to a scalar function ofcwhich, along withf, 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 thatfis locally bounded in, thatSis bounded in, and thatfor allwith someand some \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 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,. Building on this boundedness property, it can finally even be proved that in the large time limit, any such solution approaches the spatially homogeneous equilibriumin an appropriate sense, where, provided that merelyandon. 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 .