{"title":"Time-dependent flows and their applications in parabolic-parabolic Patlak-Keller-Segel systems Part I: Alternating flows","authors":"Siming He","doi":"10.1016/j.jfa.2024.110786","DOIUrl":null,"url":null,"abstract":"<div><div>We consider the three-dimensional parabolic-parabolic Patlak-Keller-Segel equations (PKS) subject to ambient flows. Without the ambient fluid flow, the equation is super-critical in three-dimension and has finite-time blow-up solutions with arbitrarily small <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-mass. In this study, we show that a family of time-dependent alternating shear flows, inspired by the clever ideas of Tarek Elgindi <span><span>[39]</span></span>, can suppress the chemotactic blow-up in these systems.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 5","pages":"Article 110786"},"PeriodicalIF":1.6000,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022123624004749","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/12/6 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the three-dimensional parabolic-parabolic Patlak-Keller-Segel equations (PKS) subject to ambient flows. Without the ambient fluid flow, the equation is super-critical in three-dimension and has finite-time blow-up solutions with arbitrarily small -mass. In this study, we show that a family of time-dependent alternating shear flows, inspired by the clever ideas of Tarek Elgindi [39], can suppress the chemotactic blow-up in these systems.
期刊介绍：
The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published.
Research Areas Include:
• Significant applications of functional analysis, including those to other areas of mathematics
• New developments in functional analysis
• Contributions to important problems in and challenges to functional analysis
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