{"title":"外平面域内Navier-Stokes系统的固定解:90年的探索、奥秘和洞见","authors":"Mikhail Korobkov, Xiao Ren","doi":"10.1007/s00021-023-00792-w","DOIUrl":null,"url":null,"abstract":"<div><p>In this survey, we study the boundary value problem for the stationary Navier–Stokes system in planar exterior domains. With no-slip boundary condition and a prescribed constant limit velocity at infinity, this problem describes stationary Navier–Stokes flows around cylindrical obstacles. Leray’s invading domains method is presented as a starting point. Then we discuss the boundedness and convergence of general <i>D</i>-solutions (solutions with finite Dirichlet integrals) in exterior domains. For the Leray solutions of the flow around an obstacle problem, we study the nontriviality, and the justification of the limit velocity at small Reynolds numbers. Further, under the same assumption of small Reynolds numbers the global uniqueness theorem for the problem is established in the class of <i>D</i>-solutions, its proof deals with the accurate perturbative analysis based on the linear Oseen system, inspired by classical Finn-Smith technique; the classical Amick and Gilbarg–Weinberger papers are involved here as well. The forced Navier–Stokes system in the whole plane is also presented as a closely related problem. A list of unsolved problems is given at the end of the paper.\n</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Stationary Solutions to the Navier–Stokes System in an Exterior Plane Domain: 90 Years of Search, Mysteries and Insights\",\"authors\":\"Mikhail Korobkov, Xiao Ren\",\"doi\":\"10.1007/s00021-023-00792-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this survey, we study the boundary value problem for the stationary Navier–Stokes system in planar exterior domains. With no-slip boundary condition and a prescribed constant limit velocity at infinity, this problem describes stationary Navier–Stokes flows around cylindrical obstacles. Leray’s invading domains method is presented as a starting point. Then we discuss the boundedness and convergence of general <i>D</i>-solutions (solutions with finite Dirichlet integrals) in exterior domains. For the Leray solutions of the flow around an obstacle problem, we study the nontriviality, and the justification of the limit velocity at small Reynolds numbers. Further, under the same assumption of small Reynolds numbers the global uniqueness theorem for the problem is established in the class of <i>D</i>-solutions, its proof deals with the accurate perturbative analysis based on the linear Oseen system, inspired by classical Finn-Smith technique; the classical Amick and Gilbarg–Weinberger papers are involved here as well. The forced Navier–Stokes system in the whole plane is also presented as a closely related problem. A list of unsolved problems is given at the end of the paper.\\n</p></div>\",\"PeriodicalId\":649,\"journal\":{\"name\":\"Journal of Mathematical Fluid Mechanics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Fluid Mechanics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00021-023-00792-w\",\"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/s00021-023-00792-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Stationary Solutions to the Navier–Stokes System in an Exterior Plane Domain: 90 Years of Search, Mysteries and Insights
In this survey, we study the boundary value problem for the stationary Navier–Stokes system in planar exterior domains. With no-slip boundary condition and a prescribed constant limit velocity at infinity, this problem describes stationary Navier–Stokes flows around cylindrical obstacles. Leray’s invading domains method is presented as a starting point. Then we discuss the boundedness and convergence of general D-solutions (solutions with finite Dirichlet integrals) in exterior domains. For the Leray solutions of the flow around an obstacle problem, we study the nontriviality, and the justification of the limit velocity at small Reynolds numbers. Further, under the same assumption of small Reynolds numbers the global uniqueness theorem for the problem is established in the class of D-solutions, its proof deals with the accurate perturbative analysis based on the linear Oseen system, inspired by classical Finn-Smith technique; the classical Amick and Gilbarg–Weinberger papers are involved here as well. The forced Navier–Stokes system in the whole plane is also presented as a closely related problem. A list of unsolved problems is given at the end of the paper.
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.