{"title":"Local Weak Solution of the Isentropic Compressible Navier–Stokes Equations with Variable Viscosity","authors":"Qin Duan, Xiangdi Huang","doi":"10.1007/s00021-024-00871-6","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we consider the 3-D compressible isentropic Navier–Stokes equations with constant shear viscosity <span>\\(\\mu \\)</span> and the bulk one <span>\\(\\lambda =b\\rho ^\\beta \\)</span>, here <i>b</i> is a positive constant, <span>\\(\\beta \\ge 0\\)</span>. This model was first introduced and well studied by Vaigant and Kazhikhov (Sib Math J 36(6):1283–1316, 1995) in 2D domain. In this paper, under the assumption that <span>\\(\\gamma >1\\)</span>, the local existence of weak solutions with higher regularity for the 3D periodic domain is established in the presence of vacuum without any smallness on the initial data. This generalize the previous paper (Desjardins in Commun Partial Differ Equ 22(5):977–1008, 1997; Huang and Yan in J Math Phys 62(11):111504, 2021) to variable viscosity coefficients. Also this is the first result concerning the local weak solution with high regularity for the Kazhikhov model in 3D case.\n</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"26 3","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-05-19","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/s00021-024-00871-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
In this paper, we consider the 3-D compressible isentropic Navier–Stokes equations with constant shear viscosity \(\mu \) and the bulk one \(\lambda =b\rho ^\beta \), here b is a positive constant, \(\beta \ge 0\). This model was first introduced and well studied by Vaigant and Kazhikhov (Sib Math J 36(6):1283–1316, 1995) in 2D domain. In this paper, under the assumption that \(\gamma >1\), the local existence of weak solutions with higher regularity for the 3D periodic domain is established in the presence of vacuum without any smallness on the initial data. This generalize the previous paper (Desjardins in Commun Partial Differ Equ 22(5):977–1008, 1997; Huang and Yan in J Math Phys 62(11):111504, 2021) to variable viscosity coefficients. Also this is the first result concerning the local weak solution with high regularity for the Kazhikhov model in 3D case.
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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