{"title":"Navier-Stokes方程的力以及Koch和Tataru定理","authors":"Pierre Gilles Lemarié-Rieusset","doi":"10.1007/s00021-023-00788-6","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the Cauchy problem for the incompressible Navier–Stokes equations on the whole space <span>\\(\\mathbb {R}^3\\)</span>, with initial value <span>\\(\\vec u_0\\in \\textrm{BMO}^{-1}\\)</span> (as in Koch and Tataru’s theorem) and with force <span>\\(\\vec f={{\\,\\textrm{div}\\,}}\\mathbb {F}\\)</span> where smallness of <span>\\(\\mathbb {F}\\)</span> ensures existence of a mild solution in absence of initial value. We study the interaction of the two solutions and discuss the existence of global solution for the complete problem (i.e. in presence of initial value and forcing term) under smallness assumptions. In particular, we discuss the interaction between Koch and Tataru solutions and Lei-Lin’s solutions (in <span>\\(L^2\\mathcal {F}^{-1}L^1\\)</span>) or solutions in the multiplier space <span>\\(\\mathcal {M}(\\dot{H}^{1/2,1}_{t,x}\\mapsto L^2_{t,x})\\)</span>.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Forces for the Navier–Stokes Equations and the Koch and Tataru Theorem\",\"authors\":\"Pierre Gilles Lemarié-Rieusset\",\"doi\":\"10.1007/s00021-023-00788-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider the Cauchy problem for the incompressible Navier–Stokes equations on the whole space <span>\\\\(\\\\mathbb {R}^3\\\\)</span>, with initial value <span>\\\\(\\\\vec u_0\\\\in \\\\textrm{BMO}^{-1}\\\\)</span> (as in Koch and Tataru’s theorem) and with force <span>\\\\(\\\\vec f={{\\\\,\\\\textrm{div}\\\\,}}\\\\mathbb {F}\\\\)</span> where smallness of <span>\\\\(\\\\mathbb {F}\\\\)</span> ensures existence of a mild solution in absence of initial value. We study the interaction of the two solutions and discuss the existence of global solution for the complete problem (i.e. in presence of initial value and forcing term) under smallness assumptions. In particular, we discuss the interaction between Koch and Tataru solutions and Lei-Lin’s solutions (in <span>\\\\(L^2\\\\mathcal {F}^{-1}L^1\\\\)</span>) or solutions in the multiplier space <span>\\\\(\\\\mathcal {M}(\\\\dot{H}^{1/2,1}_{t,x}\\\\mapsto L^2_{t,x})\\\\)</span>.</p></div>\",\"PeriodicalId\":649,\"journal\":{\"name\":\"Journal of Mathematical Fluid Mechanics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-05-25\",\"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/s00021-023-00788-6\",\"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/s00021-023-00788-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Forces for the Navier–Stokes Equations and the Koch and Tataru Theorem
We consider the Cauchy problem for the incompressible Navier–Stokes equations on the whole space \(\mathbb {R}^3\), with initial value \(\vec u_0\in \textrm{BMO}^{-1}\) (as in Koch and Tataru’s theorem) and with force \(\vec f={{\,\textrm{div}\,}}\mathbb {F}\) where smallness of \(\mathbb {F}\) ensures existence of a mild solution in absence of initial value. We study the interaction of the two solutions and discuss the existence of global solution for the complete problem (i.e. in presence of initial value and forcing term) under smallness assumptions. In particular, we discuss the interaction between Koch and Tataru solutions and Lei-Lin’s solutions (in \(L^2\mathcal {F}^{-1}L^1\)) or solutions in the multiplier space \(\mathcal {M}(\dot{H}^{1/2,1}_{t,x}\mapsto L^2_{t,x})\).
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.