The Well Posedness for Nonhomogeneous Boussinesq Equations
This paper is devoted to studying the Cauchy problem for non-homogeneous Boussinesq equations. We built the results on the critical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow>...
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| Format: | Article |
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MDPI AG
2021-11-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/11/2110 |
| _version_ | 1797508363677859840 |
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| author | Yan Liu Baiping Ouyang |
| author_facet | Yan Liu Baiping Ouyang |
| author_sort | Yan Liu |
| collection | DOAJ |
| description | This paper is devoted to studying the Cauchy problem for non-homogeneous Boussinesq equations. We built the results on the critical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>∈</mo><msubsup><mi>L</mi><mi>T</mi><mo>∞</mo></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi></mrow></msubsup><mo>)</mo></mrow><mo>×</mo><msubsup><mi>L</mi><mi>T</mi><mo>∞</mo></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mo>⋂</mo><msubsup><mi>L</mi><mi>T</mi><mn>1</mn></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn><mi>N</mi></mrow></semantics></math></inline-formula>. We proved the global existence of the solution when the initial velocity is small with respect to the viscosity, as well as the initial temperature approaches a positive constant. Furthermore, we proved the uniqueness for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo>≤</mo><mi>N</mi></mrow></semantics></math></inline-formula>. Our results can been seen as a version of symmetry in Besov space for the Boussinesq equations. |
| first_indexed | 2024-03-10T05:01:05Z |
| format | Article |
| id | doaj.art-4fef51d036ba43dda3c7a13d1c5e8ac2 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T05:01:05Z |
| publishDate | 2021-11-01 |
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| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-4fef51d036ba43dda3c7a13d1c5e8ac22023-11-23T01:45:07ZengMDPI AGSymmetry2073-89942021-11-011311211010.3390/sym13112110The Well Posedness for Nonhomogeneous Boussinesq EquationsYan Liu0Baiping Ouyang1Department of Mathematics, Guangdong University of Finance, Guangzhou 510521, ChinaSchool of Data Science, Guangzhou Huashang College, Guangzhou 511300, ChinaThis paper is devoted to studying the Cauchy problem for non-homogeneous Boussinesq equations. We built the results on the critical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>∈</mo><msubsup><mi>L</mi><mi>T</mi><mo>∞</mo></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi></mrow></msubsup><mo>)</mo></mrow><mo>×</mo><msubsup><mi>L</mi><mi>T</mi><mo>∞</mo></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mo>⋂</mo><msubsup><mi>L</mi><mi>T</mi><mn>1</mn></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn><mi>N</mi></mrow></semantics></math></inline-formula>. We proved the global existence of the solution when the initial velocity is small with respect to the viscosity, as well as the initial temperature approaches a positive constant. Furthermore, we proved the uniqueness for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo>≤</mo><mi>N</mi></mrow></semantics></math></inline-formula>. Our results can been seen as a version of symmetry in Besov space for the Boussinesq equations.https://www.mdpi.com/2073-8994/13/11/2110non homogenous boussinesq equationsglobal well-posednesslittlewood-paley decomposition |
| spellingShingle | Yan Liu Baiping Ouyang The Well Posedness for Nonhomogeneous Boussinesq Equations Symmetry non homogenous boussinesq equations global well-posedness littlewood-paley decomposition |
| title | The Well Posedness for Nonhomogeneous Boussinesq Equations |
| title_full | The Well Posedness for Nonhomogeneous Boussinesq Equations |
| title_fullStr | The Well Posedness for Nonhomogeneous Boussinesq Equations |
| title_full_unstemmed | The Well Posedness for Nonhomogeneous Boussinesq Equations |
| title_short | The Well Posedness for Nonhomogeneous Boussinesq Equations |
| title_sort | well posedness for nonhomogeneous boussinesq equations |
| topic | non homogenous boussinesq equations global well-posedness littlewood-paley decomposition |
| url | https://www.mdpi.com/2073-8994/13/11/2110 |
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