{"title":"具有质量源和强分离势的Cahn-Hilliard-Darcy体系","authors":"G. Schimperna","doi":"10.3934/dcdss.2022008","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>We study an evolutionary system of Cahn-Hilliard-Darcy type including mass source and transport effects. The system may arise in a number of physical situations related to phase separation phenomena with convection, with the main and most specific application being related to tumoral processes, where the variations of the mass may correspond to growth, or shrinking, of the tumor. We prove existence of weak solutions in the case when the configuration potential for the order parameter <inline-formula><tex-math id=\"M1\">\\begin{document}$ \\varphi $\\end{document}</tex-math></inline-formula> is designed in such a way to keep <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\varphi $\\end{document}</tex-math></inline-formula> in between the reference interval <inline-formula><tex-math id=\"M3\">\\begin{document}$ (-1, 1) $\\end{document}</tex-math></inline-formula> despite the occurrence of mass source effects. Moreover, in the two-dimensional case, we obtain existence and uniqueness of strong (i.e., more regular) solutions.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"16 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"On the Cahn-Hilliard-Darcy system with mass source and strongly separating potential\",\"authors\":\"G. Schimperna\",\"doi\":\"10.3934/dcdss.2022008\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p style='text-indent:20px;'>We study an evolutionary system of Cahn-Hilliard-Darcy type including mass source and transport effects. The system may arise in a number of physical situations related to phase separation phenomena with convection, with the main and most specific application being related to tumoral processes, where the variations of the mass may correspond to growth, or shrinking, of the tumor. We prove existence of weak solutions in the case when the configuration potential for the order parameter <inline-formula><tex-math id=\\\"M1\\\">\\\\begin{document}$ \\\\varphi $\\\\end{document}</tex-math></inline-formula> is designed in such a way to keep <inline-formula><tex-math id=\\\"M2\\\">\\\\begin{document}$ \\\\varphi $\\\\end{document}</tex-math></inline-formula> in between the reference interval <inline-formula><tex-math id=\\\"M3\\\">\\\\begin{document}$ (-1, 1) $\\\\end{document}</tex-math></inline-formula> despite the occurrence of mass source effects. Moreover, in the two-dimensional case, we obtain existence and uniqueness of strong (i.e., more regular) solutions.</p>\",\"PeriodicalId\":11254,\"journal\":{\"name\":\"Discrete & Continuous Dynamical Systems - S\",\"volume\":\"16 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-02-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete & Continuous Dynamical Systems - S\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3934/dcdss.2022008\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Continuous Dynamical Systems - S","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/dcdss.2022008","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 4
摘要
We study an evolutionary system of Cahn-Hilliard-Darcy type including mass source and transport effects. The system may arise in a number of physical situations related to phase separation phenomena with convection, with the main and most specific application being related to tumoral processes, where the variations of the mass may correspond to growth, or shrinking, of the tumor. We prove existence of weak solutions in the case when the configuration potential for the order parameter \begin{document}$ \varphi $\end{document} is designed in such a way to keep \begin{document}$ \varphi $\end{document} in between the reference interval \begin{document}$ (-1, 1) $\end{document} despite the occurrence of mass source effects. Moreover, in the two-dimensional case, we obtain existence and uniqueness of strong (i.e., more regular) solutions.
On the Cahn-Hilliard-Darcy system with mass source and strongly separating potential
We study an evolutionary system of Cahn-Hilliard-Darcy type including mass source and transport effects. The system may arise in a number of physical situations related to phase separation phenomena with convection, with the main and most specific application being related to tumoral processes, where the variations of the mass may correspond to growth, or shrinking, of the tumor. We prove existence of weak solutions in the case when the configuration potential for the order parameter \begin{document}$ \varphi $\end{document} is designed in such a way to keep \begin{document}$ \varphi $\end{document} in between the reference interval \begin{document}$ (-1, 1) $\end{document} despite the occurrence of mass source effects. Moreover, in the two-dimensional case, we obtain existence and uniqueness of strong (i.e., more regular) solutions.