{"title":"纳维-斯托克斯方程解正则性的局部标准","authors":"Congming Li , Chenkai Liu , Ran Zhuo","doi":"10.1016/j.jde.2024.09.028","DOIUrl":null,"url":null,"abstract":"<div><p>The Ladyzhenskaya-Prodi-Serrin type <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. This global <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> norm is usually large and hence hard to control. Replacing the global <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> norm with some kind of local norm is interesting. In this article, we introduce a local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> space and establish some localized criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> type norms. These local norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> type norms is necessary and sufficient to affirmatively answer the millennium problem.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A localized criterion for the regularity of solutions to Navier-Stokes equations\",\"authors\":\"Congming Li , Chenkai Liu , Ran Zhuo\",\"doi\":\"10.1016/j.jde.2024.09.028\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The Ladyzhenskaya-Prodi-Serrin type <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. This global <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> norm is usually large and hence hard to control. Replacing the global <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> norm with some kind of local norm is interesting. In this article, we introduce a local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> space and establish some localized criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> type norms. These local norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>r</mi></mrow></msup></math></span> type norms is necessary and sufficient to affirmatively answer the millennium problem.</p></div>\",\"PeriodicalId\":15623,\"journal\":{\"name\":\"Journal of Differential Equations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2024-09-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022039624006132\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039624006132","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
A localized criterion for the regularity of solutions to Navier-Stokes equations
The Ladyzhenskaya-Prodi-Serrin type criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. This global norm is usually large and hence hard to control. Replacing the global norm with some kind of local norm is interesting. In this article, we introduce a local space and establish some localized criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local type norms. These local norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local type norms is necessary and sufficient to affirmatively answer the millennium problem.
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics