Nonuniform Dependence of a Two-Component NOVIKOV System in Besov Spaces
Considered herein is the Cauchy problem of the two-component Novikov system. In the periodic case, we first constructed an approximate solution sequence that possesses the nonuniform dependence property; then, by applying the energy methods, we managed to prove that the difference between the approx...
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MDPI AG
2023-04-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/9/2041 |
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| author | Shengqi Yu Jie Liu |
| author_facet | Shengqi Yu Jie Liu |
| author_sort | Shengqi Yu |
| collection | DOAJ |
| description | Considered herein is the Cauchy problem of the two-component Novikov system. In the periodic case, we first constructed an approximate solution sequence that possesses the nonuniform dependence property; then, by applying the energy methods, we managed to prove that the difference between the approximate and actual solution is negligible, thus succeeding in proving the nonuniform dependence result in both supercritical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mi>r</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">T</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mi>r</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">T</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><mo movablelimits="true" form="prefix">max</mo><mo>{</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>}</mo><mo>,</mo><mspace width="0.166667em"></mspace></mrow></semantics></math></inline-formula><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>∞</mi><mo>,</mo><mspace width="0.166667em"></mspace><mn>1</mn><mo>≤</mo><mi>r</mi><mo><</mo><mi>∞</mi></mrow></semantics></math></inline-formula> and critical Besov space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">T</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">T</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In the non-periodic case, we constructed two sequences of initial data with high and low-frequency terms by analyzing the inner structure of the system under investigation in detail, and we proved that the distance between the two corresponding solution sequences is lower-bounded by time <i>t</i>, but converges to zero at initial time. This implies that the solution map is not uniformly continuous both in supercritical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mi>r</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mi>r</mi></mrow><mi>s</mi></msubsup><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><mo movablelimits="true" form="prefix">max</mo><mo>{</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>}</mo><mo>,</mo><mspace width="0.166667em"></mspace><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>∞</mi><mo>,</mo><mspace width="0.166667em"></mspace><mn>1</mn><mo>≤</mo><mi>r</mi><mo><</mo><mi>∞</mi></mrow></semantics></math></inline-formula> and critical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>p</mi></mfrac></mrow></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>p</mi></mfrac></mrow></msubsup><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><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula>. The proof of nonuniform dependence is based on approximate solutions and Littlewood–Paley decomposition theory. These approaches are widely applicable in the study of continuous properties for shallow water equations. |
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| language | English |
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| spelling | doaj.art-c146790b8d93461cbbd94adeaf3d463c2023-11-17T23:19:13ZengMDPI AGMathematics2227-73902023-04-01119204110.3390/math11092041Nonuniform Dependence of a Two-Component NOVIKOV System in Besov SpacesShengqi Yu0Jie Liu1School of Sciences, Nantong University, Nantong 226007, ChinaSchool of Sciences, Nantong University, Nantong 226007, ChinaConsidered herein is the Cauchy problem of the two-component Novikov system. In the periodic case, we first constructed an approximate solution sequence that possesses the nonuniform dependence property; then, by applying the energy methods, we managed to prove that the difference between the approximate and actual solution is negligible, thus succeeding in proving the nonuniform dependence result in both supercritical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mi>r</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">T</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mi>r</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">T</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><mo movablelimits="true" form="prefix">max</mo><mo>{</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>}</mo><mo>,</mo><mspace width="0.166667em"></mspace></mrow></semantics></math></inline-formula><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>∞</mi><mo>,</mo><mspace width="0.166667em"></mspace><mn>1</mn><mo>≤</mo><mi>r</mi><mo><</mo><mi>∞</mi></mrow></semantics></math></inline-formula> and critical Besov space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">T</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">T</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In the non-periodic case, we constructed two sequences of initial data with high and low-frequency terms by analyzing the inner structure of the system under investigation in detail, and we proved that the distance between the two corresponding solution sequences is lower-bounded by time <i>t</i>, but converges to zero at initial time. This implies that the solution map is not uniformly continuous both in supercritical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mi>r</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mi>r</mi></mrow><mi>s</mi></msubsup><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><mo movablelimits="true" form="prefix">max</mo><mo>{</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>}</mo><mo>,</mo><mspace width="0.166667em"></mspace><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>∞</mi><mo>,</mo><mspace width="0.166667em"></mspace><mn>1</mn><mo>≤</mo><mi>r</mi><mo><</mo><mi>∞</mi></mrow></semantics></math></inline-formula> and critical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>p</mi></mfrac></mrow></msubsup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>p</mi></mfrac></mrow></msubsup><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><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula>. The proof of nonuniform dependence is based on approximate solutions and Littlewood–Paley decomposition theory. These approaches are widely applicable in the study of continuous properties for shallow water equations.https://www.mdpi.com/2227-7390/11/9/2041two-component Novikov systemnon-uniform dependencesupercritical Besov spacescritical Besov spaces |
| spellingShingle | Shengqi Yu Jie Liu Nonuniform Dependence of a Two-Component NOVIKOV System in Besov Spaces Mathematics two-component Novikov system non-uniform dependence supercritical Besov spaces critical Besov spaces |
| title | Nonuniform Dependence of a Two-Component NOVIKOV System in Besov Spaces |
| title_full | Nonuniform Dependence of a Two-Component NOVIKOV System in Besov Spaces |
| title_fullStr | Nonuniform Dependence of a Two-Component NOVIKOV System in Besov Spaces |
| title_full_unstemmed | Nonuniform Dependence of a Two-Component NOVIKOV System in Besov Spaces |
| title_short | Nonuniform Dependence of a Two-Component NOVIKOV System in Besov Spaces |
| title_sort | nonuniform dependence of a two component novikov system in besov spaces |
| topic | two-component Novikov system non-uniform dependence supercritical Besov spaces critical Besov spaces |
| url | https://www.mdpi.com/2227-7390/11/9/2041 |
| work_keys_str_mv | AT shengqiyu nonuniformdependenceofatwocomponentnovikovsysteminbesovspaces AT jieliu nonuniformdependenceofatwocomponentnovikovsysteminbesovspaces |