{"title":"论傅立叶空间中具有流体效应的趋化系统的初始层和极限行为","authors":"Jian-xiang Wan, Hai-ping Zhong","doi":"10.1007/s10255-024-1134-3","DOIUrl":null,"url":null,"abstract":"<div><p>The paper deals with a Cauchy problem for the chemotaxis system with the effect of fluid </p><div><div><span>$$\\left\\{ {\\matrix{ {u_t^\\varepsilon + {u^\\varepsilon } \\cdot \\nabla {u^\\varepsilon } - \\Delta {u^\\varepsilon } + \\nabla {{\\rm{P}}^\\varepsilon } = {n^\\varepsilon }\\nabla {c^\\varepsilon },} \\hfill & {{\\rm{in}}} \\hfill & {{\\mathbb{R}^d} \\times \\left( {0,\\infty } \\right),} \\hfill \\cr {\\nabla \\cdot {u^\\varepsilon } = 0,} \\hfill & {{\\rm{in}}} \\hfill & {{\\mathbb{R}^d} \\times \\left( {0,\\infty } \\right),} \\hfill \\cr {n_t^\\varepsilon + {u^\\varepsilon } \\cdot \\nabla {n^\\varepsilon } - \\Delta {n^\\varepsilon } = - \\nabla \\cdot \\left( {{n^\\varepsilon }\\nabla {c^\\varepsilon }} \\right),} \\hfill & {{\\rm{in}}} \\hfill & {{\\mathbb{R}^d} \\times \\left( {0,\\infty } \\right),} \\hfill \\cr {{1 \\over \\varepsilon }c_t^\\varepsilon - \\Delta {c^\\varepsilon } = {n^\\varepsilon },} \\hfill & {{\\rm{in}}} \\hfill & {{\\mathbb{R}^d} \\times \\left( {0,\\infty } \\right),} \\hfill \\cr {\\left( {{u^\\varepsilon },{n^\\varepsilon },{c^\\varepsilon }} \\right){|_{t = 0}} = \\left( {{u_0},{n_0},{c_0}} \\right),} \\hfill & {{\\rm{in}}} \\hfill & {{\\mathbb{R}^d},} \\hfill \\cr } } \\right.$$</span></div></div><p> where <i>d</i> ≥ 2. It is known that for each <i>ϵ</i> > 0 and all sufficiently small initial data (<i>u</i><sub>0</sub>, <i>n</i><sub>0</sub>, <i>c</i><sub>0</sub>) belongs to certain Fourier space, the problem possesses a unique global solution (<i>u</i><sup><i>ϵ</i></sup>, <i>n</i><sup><i>ϵ</i></sup>, <i>c</i><sup><i>ϵ</i></sup>) in Fourier space. The present work asserts that these solutions stabilize to (<i>u</i><sup>∞</sup>, <i>n</i><sup>∞</sup>, <i>c</i><sup>∞</sup>) as <i>ϵ</i><sup>−1</sup> → 0. Moreover, we show that <i>c</i><sup><i>ϵ</i></sup>(<i>t</i>) has the initial layer as <i>ϵ</i><sup>−1</sup> → 0. As one expects its limit behavior maybe give a new viewlook to understand the system.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10255-024-1134-3.pdf","citationCount":"0","resultStr":"{\"title\":\"On the Initial Layer and the Limit Behavior for Chemotaxis System with the Effect of Fluid in Fourier Space\",\"authors\":\"Jian-xiang Wan, Hai-ping Zhong\",\"doi\":\"10.1007/s10255-024-1134-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The paper deals with a Cauchy problem for the chemotaxis system with the effect of fluid </p><div><div><span>$$\\\\left\\\\{ {\\\\matrix{ {u_t^\\\\varepsilon + {u^\\\\varepsilon } \\\\cdot \\\\nabla {u^\\\\varepsilon } - \\\\Delta {u^\\\\varepsilon } + \\\\nabla {{\\\\rm{P}}^\\\\varepsilon } = {n^\\\\varepsilon }\\\\nabla {c^\\\\varepsilon },} \\\\hfill & {{\\\\rm{in}}} \\\\hfill & {{\\\\mathbb{R}^d} \\\\times \\\\left( {0,\\\\infty } \\\\right),} \\\\hfill \\\\cr {\\\\nabla \\\\cdot {u^\\\\varepsilon } = 0,} \\\\hfill & {{\\\\rm{in}}} \\\\hfill & {{\\\\mathbb{R}^d} \\\\times \\\\left( {0,\\\\infty } \\\\right),} \\\\hfill \\\\cr {n_t^\\\\varepsilon + {u^\\\\varepsilon } \\\\cdot \\\\nabla {n^\\\\varepsilon } - \\\\Delta {n^\\\\varepsilon } = - \\\\nabla \\\\cdot \\\\left( {{n^\\\\varepsilon }\\\\nabla {c^\\\\varepsilon }} \\\\right),} \\\\hfill & {{\\\\rm{in}}} \\\\hfill & {{\\\\mathbb{R}^d} \\\\times \\\\left( {0,\\\\infty } \\\\right),} \\\\hfill \\\\cr {{1 \\\\over \\\\varepsilon }c_t^\\\\varepsilon - \\\\Delta {c^\\\\varepsilon } = {n^\\\\varepsilon },} \\\\hfill & {{\\\\rm{in}}} \\\\hfill & {{\\\\mathbb{R}^d} \\\\times \\\\left( {0,\\\\infty } \\\\right),} \\\\hfill \\\\cr {\\\\left( {{u^\\\\varepsilon },{n^\\\\varepsilon },{c^\\\\varepsilon }} \\\\right){|_{t = 0}} = \\\\left( {{u_0},{n_0},{c_0}} \\\\right),} \\\\hfill & {{\\\\rm{in}}} \\\\hfill & {{\\\\mathbb{R}^d},} \\\\hfill \\\\cr } } \\\\right.$$</span></div></div><p> where <i>d</i> ≥ 2. It is known that for each <i>ϵ</i> > 0 and all sufficiently small initial data (<i>u</i><sub>0</sub>, <i>n</i><sub>0</sub>, <i>c</i><sub>0</sub>) belongs to certain Fourier space, the problem possesses a unique global solution (<i>u</i><sup><i>ϵ</i></sup>, <i>n</i><sup><i>ϵ</i></sup>, <i>c</i><sup><i>ϵ</i></sup>) in Fourier space. The present work asserts that these solutions stabilize to (<i>u</i><sup>∞</sup>, <i>n</i><sup>∞</sup>, <i>c</i><sup>∞</sup>) as <i>ϵ</i><sup>−1</sup> → 0. Moreover, we show that <i>c</i><sup><i>ϵ</i></sup>(<i>t</i>) has the initial layer as <i>ϵ</i><sup>−1</sup> → 0. As one expects its limit behavior maybe give a new viewlook to understand the system.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-11-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s10255-024-1134-3.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10255-024-1134-3\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10255-024-1134-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
where d ≥ 2. It is known that for each ϵ > 0 and all sufficiently small initial data (u0, n0, c0) belongs to certain Fourier space, the problem possesses a unique global solution (uϵ, nϵ, cϵ) in Fourier space. The present work asserts that these solutions stabilize to (u∞, n∞, c∞) as ϵ−1 → 0. Moreover, we show that cϵ(t) has the initial layer as ϵ−1 → 0. As one expects its limit behavior maybe give a new viewlook to understand the system.