{"title":"Gevrey Type Error Estimates of Solutions to the Navier–Stokes Equations","authors":"Yuta Koizumi","doi":"10.1007/s00021-025-00924-4","DOIUrl":null,"url":null,"abstract":"<div><p>Consider the Cauchy problem of the Navier–Stokes equations in <span>\\(\\mathbb {R}^n (n \\ge 2)\\)</span> with the initial data <span>\\(a \\in \\dot{B}^{-1+n/p}_{p, \\infty }\\)</span> for <span>\\(n< p < \\infty \\)</span>. We establish the Gevrey type estimates for the error between the successive approximations <span>\\(\\{u_j\\}_{j=0}^{\\infty }\\)</span> and the strong solution <i>u</i> provided the convergence in the scaling invariant norm in <span>\\(L^q(\\mathbb {R}^n)\\)</span> with the time weight holds. It is also clarified that the convergence rate of the higher order approximation is at least the same as that of the lower order approximation. In addition, the approximation for the pressure is also established.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"27 2","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2025-02-07","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/s00021-025-00924-4","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
Consider the Cauchy problem of the Navier–Stokes equations in \(\mathbb {R}^n (n \ge 2)\) with the initial data \(a \in \dot{B}^{-1+n/p}_{p, \infty }\) for \(n< p < \infty \). We establish the Gevrey type estimates for the error between the successive approximations \(\{u_j\}_{j=0}^{\infty }\) and the strong solution u provided the convergence in the scaling invariant norm in \(L^q(\mathbb {R}^n)\) with the time weight holds. It is also clarified that the convergence rate of the higher order approximation is at least the same as that of the lower order approximation. In addition, the approximation for the pressure is also established.
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.