{"title":"Smooth solutions in a three-dimensional chemotaxis-Stokes system involving Dirichlet boundary conditions for the signal","authors":"Yulan Wang, Michael Winkler, Zhaoyin Xiang","doi":"10.1007/s00030-024-00982-z","DOIUrl":null,"url":null,"abstract":"<p>In a smoothly bounded domain <span>\\(\\Omega \\subset \\mathbb {R}^3\\)</span>, the chemotaxis-Stokes system </p><span>$$\\begin{aligned} \\left\\{ \\begin{array}{l} n_t + u\\cdot \\nabla n = \\Delta n - \\nabla \\cdot (n\\nabla c), \\\\ c_t + u\\cdot \\nabla c =\\Delta c - nc, \\\\ u_t = \\Delta u + \\nabla P + n\\nabla \\phi , \\qquad \\nabla \\cdot u =0 \\end{array} \\right. \\end{aligned}$$</span><p>is considered along with the boundary conditions </p><span>$$\\begin{aligned} \\big (\\nabla n - n\\nabla c\\big )\\cdot \\nu = 0, \\quad c=c_\\star , \\quad u=0, \\quad x\\in \\partial \\Omega , \\,\\, t>0, \\end{aligned}$$</span><p>where <span>\\(c_\\star \\ge 0\\)</span> is a given constant. It is shown that under a smallness condition on <span>\\(c(\\cdot ,0)\\)</span> and suitable assumptions on regularity of the initial data, global classical solutions exist which are uniformly bounded.\n</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"42 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Differential Equations and Applications (NoDEA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00030-024-00982-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In a smoothly bounded domain \(\Omega \subset \mathbb {R}^3\), the chemotaxis-Stokes system
$$\begin{aligned} \left\{ \begin{array}{l} n_t + u\cdot \nabla n = \Delta n - \nabla \cdot (n\nabla c), \\ c_t + u\cdot \nabla c =\Delta c - nc, \\ u_t = \Delta u + \nabla P + n\nabla \phi , \qquad \nabla \cdot u =0 \end{array} \right. \end{aligned}$$
where \(c_\star \ge 0\) is a given constant. It is shown that under a smallness condition on \(c(\cdot ,0)\) and suitable assumptions on regularity of the initial data, global classical solutions exist which are uniformly bounded.