{"title":"Inhomogenous Navier–Stokes Equations with Unbounded Density","authors":"Jean-Paul Adogbo, Piotr B. Mucha, Maja Szlenk","doi":"10.1007/s00021-025-00956-w","DOIUrl":null,"url":null,"abstract":"<div><p>In the current state of the art regarding the Navier–Stokes equations, the existence of unique solutions for incompressible flows in two spatial dimensions is already well-established. Recently, these results have been extended to models with variable density, maintaining positive outcomes for merely bounded densities, even in cases with large vacuum regions. However, the study of incompressible Navier-Stokes equations with unbounded densities remains incomplete. Addressing this gap is the focus of the present paper. Our main result demonstrates the global existence of a unique solution for flows initiated by unbounded density, whose regularity/integrability is characterized within a specific subset of the Yudovich class of unbounded functions. The core of our proof lies in the application of Desjardins’ inequality, combined with a blow-up criterion for ordinary differential equations. Furthermore, we derive time-weighted estimates that guarantee the existence of a <span>\\(C^1\\)</span> velocity field and ensure the equivalence of Eulerian and Lagrangian formulations of the equations. Finally, by leveraging results from Danchin, R., Mucha, P.B.: The incompressible Navier-Stokes equations in vacuum. Comm. Pure Appl. Math <b>72</b>(7), 1351–1385 (2019), we conclude the uniqueness of the solution.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"27 3","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-025-00956-w.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Fluid Mechanics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00021-025-00956-w","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
In the current state of the art regarding the Navier–Stokes equations, the existence of unique solutions for incompressible flows in two spatial dimensions is already well-established. Recently, these results have been extended to models with variable density, maintaining positive outcomes for merely bounded densities, even in cases with large vacuum regions. However, the study of incompressible Navier-Stokes equations with unbounded densities remains incomplete. Addressing this gap is the focus of the present paper. Our main result demonstrates the global existence of a unique solution for flows initiated by unbounded density, whose regularity/integrability is characterized within a specific subset of the Yudovich class of unbounded functions. The core of our proof lies in the application of Desjardins’ inequality, combined with a blow-up criterion for ordinary differential equations. Furthermore, we derive time-weighted estimates that guarantee the existence of a \(C^1\) velocity field and ensure the equivalence of Eulerian and Lagrangian formulations of the equations. Finally, by leveraging results from Danchin, R., Mucha, P.B.: The incompressible Navier-Stokes equations in vacuum. Comm. Pure Appl. Math 72(7), 1351–1385 (2019), we conclude the uniqueness of the solution.
在目前关于Navier-Stokes方程的研究中,二维不可压缩流的唯一解的存在性已经得到了证实。最近,这些结果已扩展到具有可变密度的模型,即使在具有大真空区域的情况下,仅在有界密度的情况下也保持积极的结果。然而,具有无界密度的不可压缩Navier-Stokes方程的研究仍然不完整。解决这一差距是本文的重点。我们的主要结果证明了由无界密度引发的流动的一个唯一解的整体存在性,其正则性/可积性在无界函数的Yudovich类的一个特定子集内表征。我们证明的核心在于Desjardins不等式的应用,并结合常微分方程的膨胀判据。进一步,我们推导了时间加权估计,保证了\(C^1\)速度场的存在性,并保证了方程的欧拉式和拉格朗日式的等价性。最后,通过利用Danchin, R., Mucha, p.b.的结果:真空中不可压缩的Navier-Stokes方程。纯苹果通讯公司。数学72(7),1351-1385(2019),我们得出解的唯一性。
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.