{"title":"Global Bounded Solution in a Chemotaxis-Stokes Model with Porous Medium Diffusion and Singular Sensitivity","authors":"Jianping Wang","doi":"10.1007/s10440-023-00599-x","DOIUrl":null,"url":null,"abstract":"<div><p>This article is concerned with a chemotaxis-Stokes system with porous medium diffusion and singular sensitivity: </p><div><div><span> $$\\begin{aligned} \\left \\{ \\textstyle\\begin{array}{l@{\\quad }l} n_{t}+u\\cdot \\nabla n=\\nabla \\cdot (D(n)\\nabla n)-\\nabla \\cdot (nS(x,n,c)\\cdot \\nabla c),&x\\in \\Omega ,\\ \\ t>0, \\\\ c_{t}+u\\cdot \\nabla c=\\Delta c-nc,&x\\in \\Omega ,\\ \\ t>0, \\\\ u_{t}+\\nabla P=\\Delta u+n\\nabla \\Phi ,\\ \\ \\ \\nabla \\cdot u=0,&x\\in \\Omega ,\\ \\ t>0 \\end{array}\\displaystyle \\right . \\end{aligned}$$ </span></div></div><p> in a bounded domain <span>\\(\\Omega \\subset \\mathbb{R}^{N}\\)</span> with <span>\\(2\\le N\\le 3\\)</span>, where <span>\\(D\\in C^{0}([0,\\infty ))\\cap C^{2}((0,\\infty ))\\)</span> and <span>\\(S\\in C^{2}(\\bar{\\Omega }\\times [0,\\infty )^{2};\\mathbb{R}^{N\\times N})\\)</span>. The global solvability of the system in a natural weak sense is obtained under the conditions that <span>\\(D(n)\\ge k_{D}n^{m-1}\\)</span> and <span>\\(|S(x,n,c)|\\le \\frac{S_{0}(c)}{c^{\\alpha }}\\)</span> for all <span>\\((x,n,c)\\in \\Omega \\times (0,\\infty )^{2}\\)</span> with some <span>\\(k_{D}>0\\)</span>, <span>\\(m>\\frac{3N-2}{2N}\\)</span>, <span>\\(\\alpha \\in [0,1)\\)</span> and some nondecreasing <span>\\(S_{0}:(0,\\infty )\\rightarrow (0,\\infty )\\)</span>. Moreover, in the case that <span>\\(m=\\frac{3N-2}{2N}\\)</span> and <span>\\(\\alpha \\in [0,1)\\)</span>, we also get the global weak solutions under smallness assumptions on the initial data <span>\\(\\|n_{0}\\|_{L^{1}(\\Omega )}\\)</span> and <span>\\(\\|c_{0}\\|_{L^{\\infty }(\\Omega )}\\)</span>.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10440-023-00599-x.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Applicandae Mathematicae","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10440-023-00599-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This article is concerned with a chemotaxis-Stokes system with porous medium diffusion and singular sensitivity:
in a bounded domain \(\Omega \subset \mathbb{R}^{N}\) with \(2\le N\le 3\), where \(D\in C^{0}([0,\infty ))\cap C^{2}((0,\infty ))\) and \(S\in C^{2}(\bar{\Omega }\times [0,\infty )^{2};\mathbb{R}^{N\times N})\). The global solvability of the system in a natural weak sense is obtained under the conditions that \(D(n)\ge k_{D}n^{m-1}\) and \(|S(x,n,c)|\le \frac{S_{0}(c)}{c^{\alpha }}\) for all \((x,n,c)\in \Omega \times (0,\infty )^{2}\) with some \(k_{D}>0\), \(m>\frac{3N-2}{2N}\), \(\alpha \in [0,1)\) and some nondecreasing \(S_{0}:(0,\infty )\rightarrow (0,\infty )\). Moreover, in the case that \(m=\frac{3N-2}{2N}\) and \(\alpha \in [0,1)\), we also get the global weak solutions under smallness assumptions on the initial data \(\|n_{0}\|_{L^{1}(\Omega )}\) and \(\|c_{0}\|_{L^{\infty }(\Omega )}\).
期刊介绍:
Acta Applicandae Mathematicae is devoted to the art and techniques of applying mathematics and the development of new, applicable mathematical methods.
Covering a large spectrum from modeling to qualitative analysis and computational methods, Acta Applicandae Mathematicae contains papers on different aspects of the relationship between theory and applications, ranging from descriptive papers on actual applications meeting contemporary mathematical standards to proofs of new and deep theorems in applied mathematics.