Continuity and Analyticity for the Generalized Benjamin–Ono Equation
This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such dependence is no...
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MDPI AG
2021-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/12/2435 |
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| author | Xiaolin Pan Bin Wang Rong Chen |
| author_facet | Xiaolin Pan Bin Wang Rong Chen |
| author_sort | Xiaolin Pan |
| collection | DOAJ |
| description | This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>s</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>></mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>. We also provide more information about the stability of the data-solution map, i.e., the solution map for g-BO equation is Hölder continuous in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mi>r</mi></msup></semantics></math></inline-formula>-topology for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>r</mi><mo><</mo><mi>s</mi></mrow></semantics></math></inline-formula> with exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> depending on <i>s</i> and <i>r</i>. Finally, applying the generalized Ovsyannikov type theorem and the basic properties of Sobolev–Gevrey spaces, we prove the Gevrey regularity and analyticity for the g-BO equation. In addition, by the symmetry of the spatial variable, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map. |
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| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T03:59:42Z |
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| spelling | doaj.art-275b5587839144b1894d1f3b43bd651b2023-11-23T10:47:12ZengMDPI AGSymmetry2073-89942021-12-011312243510.3390/sym13122435Continuity and Analyticity for the Generalized Benjamin–Ono EquationXiaolin Pan0Bin Wang1Rong Chen2College of Mathematics Science, Chongqing Normal University, Chongqing 401331, ChinaChongqing Fengmingshan High School, Chongqing 401331, ChinaPersonnel Department, Chongqing Normal University, Chongqing 401331, ChinaThis work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>s</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>></mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>. We also provide more information about the stability of the data-solution map, i.e., the solution map for g-BO equation is Hölder continuous in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mi>r</mi></msup></semantics></math></inline-formula>-topology for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>r</mi><mo><</mo><mi>s</mi></mrow></semantics></math></inline-formula> with exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> depending on <i>s</i> and <i>r</i>. Finally, applying the generalized Ovsyannikov type theorem and the basic properties of Sobolev–Gevrey spaces, we prove the Gevrey regularity and analyticity for the g-BO equation. In addition, by the symmetry of the spatial variable, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map.https://www.mdpi.com/2073-8994/13/12/2435generalized Benjamin–Ono equationnon-uniform dependenceHölder continuoussymmetryanalyticityGevrey regularity |
| spellingShingle | Xiaolin Pan Bin Wang Rong Chen Continuity and Analyticity for the Generalized Benjamin–Ono Equation Symmetry generalized Benjamin–Ono equation non-uniform dependence Hölder continuous symmetry analyticity Gevrey regularity |
| title | Continuity and Analyticity for the Generalized Benjamin–Ono Equation |
| title_full | Continuity and Analyticity for the Generalized Benjamin–Ono Equation |
| title_fullStr | Continuity and Analyticity for the Generalized Benjamin–Ono Equation |
| title_full_unstemmed | Continuity and Analyticity for the Generalized Benjamin–Ono Equation |
| title_short | Continuity and Analyticity for the Generalized Benjamin–Ono Equation |
| title_sort | continuity and analyticity for the generalized benjamin ono equation |
| topic | generalized Benjamin–Ono equation non-uniform dependence Hölder continuous symmetry analyticity Gevrey regularity |
| url | https://www.mdpi.com/2073-8994/13/12/2435 |
| work_keys_str_mv | AT xiaolinpan continuityandanalyticityforthegeneralizedbenjaminonoequation AT binwang continuityandanalyticityforthegeneralizedbenjaminonoequation AT rongchen continuityandanalyticityforthegeneralizedbenjaminonoequation |