{"title":"Global Existence and Decaying Rates of the Strong Solution for the Boussinesq System","authors":"Lu Wang, Shuokai Yan, Qinghua Zhang","doi":"10.1155/2023/6512823","DOIUrl":null,"url":null,"abstract":"<jats:p>This paper focuses on the global existence and time-decay rates of the strong solution for the Boussinesq system with full viscosity in <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\n <msup>\n <mrow>\n <mi mathvariant=\"double-struck\">R</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> for <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\n <mi>n</mi>\n <mo>≥</mo>\n <mn>3</mn>\n </math>\n </jats:inline-formula>. Under the initial assumption of <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <msub>\n <mrow>\n <mi>θ</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>u</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n </mrow>\n </mfenced>\n <mo>∈</mo>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n <mo>/</mo>\n <mn>3</mn>\n </mrow>\n </msup>\n <mo>×</mo>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <mi>n</mi>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> with a small norm, and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\n <mi>n</mi>\n <mo>></mo>\n <mn>3</mn>\n </math>\n </jats:inline-formula> or <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\n <mi>n</mi>\n <mo>=</mo>\n <mn>3</mn>\n </math>\n </jats:inline-formula> and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\">\n <msub>\n <mrow>\n <mi>θ</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>∈</mo>\n <msup>\n <mrow>\n <mi>L</mi>\n </mrow>\n <mrow>\n <msub>\n <mrow>\n <mi>r</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> for some <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\">\n <msub>\n <mrow>\n <mi>r</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n </mrow>\n </msub>\n <mo>></mo>\n <mn>1</mn>\n </math>\n </jats:inline-formula>, global existence and uniqueness of the strong solution <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M8\">\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>θ</mi>\n <mo>,</mo>\n <mi>u</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> for the Boussinesq system is established. This solution is proven to obey the following estimates: <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M9\">\n <msub>\n <mrow>\n <mfenced open=\"‖\" close=\"‖\" separators=\"|\">\n <mrow>\n <mi>θ</mi>\n <mrow>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>t</mi>\n </mrow>\n </mfenced>\n </mrow>\n </mrow>\n </mfenced>\n </mrow>\n <mrow>\n <mi>r</mi>\n </mrow>\n </msub>\n <mo>≤</mo>\n <mi>C</mi>\n <msup>\n <mrow>\n <mi>t</mi>\n </mrow>\n <mrow>\n <mo>−</mo>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mn>3</mn>\n <mo>−</mo>\n <mi>n</mi>\n <mo>/</mo>\n <mi>p</mi>\n </mrow>\n </mfenced>\n <mo>/</mo>\n <mn>2</mn>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> for <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M10\">\n <mi>n</mi>\n <mo>/</mo>\n <mn>3</mn>\n <mo>≤</mo>\n <mi>p</mi>\n <mo><</mo>\n <mi>∞</mi>\n </math>\n </jats:inline-formula>, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M11\">\n <msub>\n <mrow>\n <mfenced open=\"‖\" close=\"‖\" separators=\"|\">\n <mrow>\n <mi>u</mi>\n <mrow>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>t</mi>\n </mrow>\n </mfenced>\n </mrow>\n </mrow>\n </mfenced>\n </mrow>\n <mrow>\n <mi>p</mi>\n </mrow>\n </msub>\n <mo>≤</mo>\n <mi>C</mi>\n <msup>\n <mrow>\n <mi>t</mi>\n </mrow>\n <mrow>\n <mo>−</mo>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mn>1</mn>\n <mo>−</mo>\n <mi>n</mi>\n <mo>/</mo>\n <mi>q</mi>\n </mrow>\n </mfenced>\n <mo>/</mo>\n <mn>2</mn>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> for <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M12\">\n <mi>n</mi>\n <mo>≤</mo>\n <mi>q</mi>\n <mo>≤</mo>\n <mi>∞</mi>\n ","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Muenster Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2023/6512823","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper focuses on the global existence and time-decay rates of the strong solution for the Boussinesq system with full viscosity in for . Under the initial assumption of with a small norm, and or and for some , global existence and uniqueness of the strong solution for the Boussinesq system is established. This solution is proven to obey the following estimates: for , for