{"title":"磁流体动力学方程极弱解的正则性","authors":"Baishun Lai, Ge Tang","doi":"10.1007/s00021-023-00805-8","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, employing the duality technique, we prove that the very weak solution of Magneto-Hydrodynamics equations is regular in <span>\\(\\mathbb {R}^3\\times (0, T]\\)</span> if it belongs to the Banach space <span>\\(L^{p}(h,T;L^{q}(\\mathbb {R}^{3}))\\)</span> with <span>\\( \\frac{2}{p}+\\frac{3}{q}=1,\\ \\ q\\in (3,\\infty )\\)</span> for any small <span>\\(h>0\\)</span>. Secondly, we further prove the integrability condition imposed on the magnetic field can be removed by using the energy method and the regularity theory of the heat operator, which is of independent interest.\n</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00021-023-00805-8.pdf","citationCount":"0","resultStr":"{\"title\":\"The Regularity of Very Weak Solutions to Magneto-Hydrodynamics Equations\",\"authors\":\"Baishun Lai, Ge Tang\",\"doi\":\"10.1007/s00021-023-00805-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, employing the duality technique, we prove that the very weak solution of Magneto-Hydrodynamics equations is regular in <span>\\\\(\\\\mathbb {R}^3\\\\times (0, T]\\\\)</span> if it belongs to the Banach space <span>\\\\(L^{p}(h,T;L^{q}(\\\\mathbb {R}^{3}))\\\\)</span> with <span>\\\\( \\\\frac{2}{p}+\\\\frac{3}{q}=1,\\\\ \\\\ q\\\\in (3,\\\\infty )\\\\)</span> for any small <span>\\\\(h>0\\\\)</span>. Secondly, we further prove the integrability condition imposed on the magnetic field can be removed by using the energy method and the regularity theory of the heat operator, which is of independent interest.\\n</p></div>\",\"PeriodicalId\":649,\"journal\":{\"name\":\"Journal of Mathematical Fluid Mechanics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-05-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00021-023-00805-8.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/s00021-023-00805-8\",\"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/s00021-023-00805-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
The Regularity of Very Weak Solutions to Magneto-Hydrodynamics Equations
In this paper, employing the duality technique, we prove that the very weak solution of Magneto-Hydrodynamics equations is regular in \(\mathbb {R}^3\times (0, T]\) if it belongs to the Banach space \(L^{p}(h,T;L^{q}(\mathbb {R}^{3}))\) with \( \frac{2}{p}+\frac{3}{q}=1,\ \ q\in (3,\infty )\) for any small \(h>0\). Secondly, we further prove the integrability condition imposed on the magnetic field can be removed by using the energy method and the regularity theory of the heat operator, which is of independent interest.
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.