{"title":"The Navier–Stokes Cauchy Problem in a Class of Weighted Function Spaces","authors":"Paolo Maremonti, Vittorio Pane","doi":"10.1007/s00021-025-00923-5","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the Navier–Stokes Cauchy problem with an initial datum in a weighted Lebesgue space. The weight is a radial function increasing at infinity. Our study partially follows the ideas of the paper by Galdi and Maremonti (J Math Fluid Mech 25:7, 2023). The authors of the quoted paper consider a special study of stability of steady fluid motions. The results hold in 3D and for small data. Here, relatively to the perturbations of the rest state, we generalize the result. We study the <i>n</i>D Navier–Stokes Cauchy problem, <span>\\(n\\ge 3\\)</span>. We prove the existence (local) of a unique regular solution. Moreover, the solution enjoys a spatial asymptotic decay whose order of decay is connected to the weight.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"27 2","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2025-02-20","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/s00021-025-00923-5","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the Navier–Stokes Cauchy problem with an initial datum in a weighted Lebesgue space. The weight is a radial function increasing at infinity. Our study partially follows the ideas of the paper by Galdi and Maremonti (J Math Fluid Mech 25:7, 2023). The authors of the quoted paper consider a special study of stability of steady fluid motions. The results hold in 3D and for small data. Here, relatively to the perturbations of the rest state, we generalize the result. We study the nD Navier–Stokes Cauchy problem, \(n\ge 3\). We prove the existence (local) of a unique regular solution. Moreover, the solution enjoys a spatial asymptotic decay whose order of decay is connected to the weight.
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.
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