{"title":"黏度随密度变化的可压缩双流体模型的适定性结果","authors":"Sagbo Marcel Zodji","doi":"10.1007/s00021-025-00954-y","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study a system of PDEs describing the motion of two compressible viscous fluids occupying the whole space <span>\\(\\mathbb {R}^d,\\;(d\\in \\{2,3\\}\\)</span>. The two phases of the mixture are separated by a <span>\\({\\mathscr {C}}^{1+\\alpha }\\)</span>-regular sharp interface <span>\\({\\mathcal {C}}\\)</span> across which the density can experience jumps. We prove the existence of a unique local-in-time solution assuming that the initial density is <span>\\(\\alpha \\)</span>-Hölder continuous on both sides of <span>\\({\\mathcal {C}}\\)</span>. The initial velocity belongs to the Sobolev space <span>\\(H^1(\\mathbb {R}^d)\\)</span>, and the divergence of the initial stress tensor belongs to <span>\\(L^2(\\mathbb {R}^d)\\)</span>. The later assumption expresses somehow the continuity of the normal component of the stress tensor. This result is more general than the one by Tani [Two-phase free boundary problem for compressible viscous fluid motion. Journal of Mathematics of Kyoto University 24(2): 243–267, 1984] as it allows for less regular initial data and furthermore it can serve as a building block for the construction of global-in-time solutions.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"27 3","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Well-Posedness Result for the Compressible Two-Fluid Model with Density-Dependent Viscosity\",\"authors\":\"Sagbo Marcel Zodji\",\"doi\":\"10.1007/s00021-025-00954-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we study a system of PDEs describing the motion of two compressible viscous fluids occupying the whole space <span>\\\\(\\\\mathbb {R}^d,\\\\;(d\\\\in \\\\{2,3\\\\}\\\\)</span>. The two phases of the mixture are separated by a <span>\\\\({\\\\mathscr {C}}^{1+\\\\alpha }\\\\)</span>-regular sharp interface <span>\\\\({\\\\mathcal {C}}\\\\)</span> across which the density can experience jumps. We prove the existence of a unique local-in-time solution assuming that the initial density is <span>\\\\(\\\\alpha \\\\)</span>-Hölder continuous on both sides of <span>\\\\({\\\\mathcal {C}}\\\\)</span>. The initial velocity belongs to the Sobolev space <span>\\\\(H^1(\\\\mathbb {R}^d)\\\\)</span>, and the divergence of the initial stress tensor belongs to <span>\\\\(L^2(\\\\mathbb {R}^d)\\\\)</span>. The later assumption expresses somehow the continuity of the normal component of the stress tensor. This result is more general than the one by Tani [Two-phase free boundary problem for compressible viscous fluid motion. Journal of Mathematics of Kyoto University 24(2): 243–267, 1984] as it allows for less regular initial data and furthermore it can serve as a building block for the construction of global-in-time solutions.</p></div>\",\"PeriodicalId\":649,\"journal\":{\"name\":\"Journal of Mathematical Fluid Mechanics\",\"volume\":\"27 3\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2025-06-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Fluid Mechanics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00021-025-00954-y\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Fluid Mechanics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00021-025-00954-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
A Well-Posedness Result for the Compressible Two-Fluid Model with Density-Dependent Viscosity
In this paper, we study a system of PDEs describing the motion of two compressible viscous fluids occupying the whole space \(\mathbb {R}^d,\;(d\in \{2,3\}\). The two phases of the mixture are separated by a \({\mathscr {C}}^{1+\alpha }\)-regular sharp interface \({\mathcal {C}}\) across which the density can experience jumps. We prove the existence of a unique local-in-time solution assuming that the initial density is \(\alpha \)-Hölder continuous on both sides of \({\mathcal {C}}\). The initial velocity belongs to the Sobolev space \(H^1(\mathbb {R}^d)\), and the divergence of the initial stress tensor belongs to \(L^2(\mathbb {R}^d)\). The later assumption expresses somehow the continuity of the normal component of the stress tensor. This result is more general than the one by Tani [Two-phase free boundary problem for compressible viscous fluid motion. Journal of Mathematics of Kyoto University 24(2): 243–267, 1984] as it allows for less regular initial data and furthermore it can serve as a building block for the construction of global-in-time solutions.
期刊介绍:
The Journal of Mathematical Fluid Mechanics (JMFM)is a forum for the publication of high-quality peer-reviewed papers on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. As an important part of that, the journal encourages papers dealing with mathematical aspects of computational theory, as well as with applications in science and engineering. The journal also publishes in related areas of mathematics that have a direct bearing on the mathematical theory of fluid mechanics. All papers will be characterized by originality and mathematical rigor. For a paper to be accepted, it is not enough that it contains original results. In fact, results should be highly relevant to the mathematical theory of fluid mechanics, and meet a wide readership.