{"title":"三维Navier-Stokes和Euler方程的分数voight正则化:全局适定性和极限行为","authors":"Zdzisław Brzeźniak, Adam Larios, Isabel Safarik","doi":"10.1007/s00021-025-00948-w","DOIUrl":null,"url":null,"abstract":"<div><p>The Voigt regularization is a technique used to model turbulent flows, offering advantages such as sharing steady states with the Navier-Stokes equations and requiring no modification of boundary conditions; however, the parabolic dissipative character of the equation is lost. In this work we propose and study a generalization of the Voigt regularization technique by introducing a fractional power <i>r</i> in the Helmholtz operator, which allows for dissipation in the system, at least in the viscous case. We examine the resulting fractional Navier-Stokes-Voigt (fNSV) and fractional Euler-Voigt (fEV) and show that global well-posedness holds in the 3D periodic case for fNSV when the fractional power <span>\\(r \\ge \\frac{1}{2}\\)</span> and for fEV when <span>\\(r>\\frac{5}{6}\\)</span>. Moreover, we show that the solutions of these fractional Voigt-regularized systems converge to solutions of the original equations, on the corresponding time interval of existence and uniqueness of the latter, as the regularization parameter <span>\\(\\alpha \\rightarrow 0\\)</span>. Additionally, we prove convergence of solutions of fNSV to solutions of fEV as the viscosity <span>\\(\\nu \\rightarrow 0\\)</span> as well as the convergence of solutions of fNSV to solutions of the 3D Euler equations as both <span>\\(\\alpha , \\nu \\rightarrow 0\\)</span>. Furthermore, we derive a criterion for finite-time blow-up for each system based on this regularization. These results may be of use to researchers in both pure and applied fluid dynamics, particularly in terms of approximate models for turbulence and as tools to investigate potential blow-up of solutions.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"27 3","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fractional Voigt-Regularization of the 3D Navier–Stokes and Euler Equations: Global Well-Posedness and Limiting Behavior\",\"authors\":\"Zdzisław Brzeźniak, Adam Larios, Isabel Safarik\",\"doi\":\"10.1007/s00021-025-00948-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The Voigt regularization is a technique used to model turbulent flows, offering advantages such as sharing steady states with the Navier-Stokes equations and requiring no modification of boundary conditions; however, the parabolic dissipative character of the equation is lost. In this work we propose and study a generalization of the Voigt regularization technique by introducing a fractional power <i>r</i> in the Helmholtz operator, which allows for dissipation in the system, at least in the viscous case. We examine the resulting fractional Navier-Stokes-Voigt (fNSV) and fractional Euler-Voigt (fEV) and show that global well-posedness holds in the 3D periodic case for fNSV when the fractional power <span>\\\\(r \\\\ge \\\\frac{1}{2}\\\\)</span> and for fEV when <span>\\\\(r>\\\\frac{5}{6}\\\\)</span>. Moreover, we show that the solutions of these fractional Voigt-regularized systems converge to solutions of the original equations, on the corresponding time interval of existence and uniqueness of the latter, as the regularization parameter <span>\\\\(\\\\alpha \\\\rightarrow 0\\\\)</span>. Additionally, we prove convergence of solutions of fNSV to solutions of fEV as the viscosity <span>\\\\(\\\\nu \\\\rightarrow 0\\\\)</span> as well as the convergence of solutions of fNSV to solutions of the 3D Euler equations as both <span>\\\\(\\\\alpha , \\\\nu \\\\rightarrow 0\\\\)</span>. Furthermore, we derive a criterion for finite-time blow-up for each system based on this regularization. These results may be of use to researchers in both pure and applied fluid dynamics, particularly in terms of approximate models for turbulence and as tools to investigate potential blow-up of solutions.</p></div>\",\"PeriodicalId\":649,\"journal\":{\"name\":\"Journal of Mathematical Fluid Mechanics\",\"volume\":\"27 3\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2025-06-09\",\"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/s00021-025-00948-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/s00021-025-00948-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Fractional Voigt-Regularization of the 3D Navier–Stokes and Euler Equations: Global Well-Posedness and Limiting Behavior
The Voigt regularization is a technique used to model turbulent flows, offering advantages such as sharing steady states with the Navier-Stokes equations and requiring no modification of boundary conditions; however, the parabolic dissipative character of the equation is lost. In this work we propose and study a generalization of the Voigt regularization technique by introducing a fractional power r in the Helmholtz operator, which allows for dissipation in the system, at least in the viscous case. We examine the resulting fractional Navier-Stokes-Voigt (fNSV) and fractional Euler-Voigt (fEV) and show that global well-posedness holds in the 3D periodic case for fNSV when the fractional power \(r \ge \frac{1}{2}\) and for fEV when \(r>\frac{5}{6}\). Moreover, we show that the solutions of these fractional Voigt-regularized systems converge to solutions of the original equations, on the corresponding time interval of existence and uniqueness of the latter, as the regularization parameter \(\alpha \rightarrow 0\). Additionally, we prove convergence of solutions of fNSV to solutions of fEV as the viscosity \(\nu \rightarrow 0\) as well as the convergence of solutions of fNSV to solutions of the 3D Euler equations as both \(\alpha , \nu \rightarrow 0\). Furthermore, we derive a criterion for finite-time blow-up for each system based on this regularization. These results may be of use to researchers in both pure and applied fluid dynamics, particularly in terms of approximate models for turbulence and as tools to investigate potential blow-up of solutions.
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.