{"title":"On the blow-up criterion for the Hall-MHD problem with partial dissipation in $\\mathbb{R}^{3}$","authors":"Baoying Du","doi":"10.1186/s13661-023-01723-4","DOIUrl":null,"url":null,"abstract":"Abstract In this paper, we investigate the 3D incompressible Hall-magnetohydrodynamics with partial dissipation. Based on the results in (Du in Bound. Value Probl. 2022:6, 2022; Du and Liu in Acta Math. Sci. 42A:5, 2022; Fei and Xiang in J. Math. Phys. 56:051504, 2015), we establish an improved blow-up criterion for classical solutions. Furthermore, using the blow-up criterion, we also obtain the existence of the classical solutions only under the condition that the initial data $\\|V_{0}\\|_{H^{1}}+\\|B_{0}\\|_{H^{2}}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mrow> <mml:mo>∥</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∥</mml:mo> </mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow> <mml:mo>∥</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∥</mml:mo> </mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:msub> </mml:math> are sufficiently small.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"12 1","pages":"0"},"PeriodicalIF":1.0000,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Boundary Value Problems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1186/s13661-023-01723-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract In this paper, we investigate the 3D incompressible Hall-magnetohydrodynamics with partial dissipation. Based on the results in (Du in Bound. Value Probl. 2022:6, 2022; Du and Liu in Acta Math. Sci. 42A:5, 2022; Fei and Xiang in J. Math. Phys. 56:051504, 2015), we establish an improved blow-up criterion for classical solutions. Furthermore, using the blow-up criterion, we also obtain the existence of the classical solutions only under the condition that the initial data $\|V_{0}\|_{H^{1}}+\|B_{0}\|_{H^{2}}$ ∥V0∥H1+∥B0∥H2 are sufficiently small.
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