{"title":"三维超耗散纳维-斯托克斯方程的尖锐非唯一性:超越狮子指数","authors":"Yachun Li , Peng Qu , Zirong Zeng , Deng Zhang","doi":"10.1016/j.matpur.2024.103602","DOIUrl":null,"url":null,"abstract":"<div><p>We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent <em>α</em> can be larger than the Lions exponent 5/4. It is well-known that, due to Lions <span><span>[1]</span></span>, for any <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> divergence-free initial data, there exist unique smooth Leray-Hopf solutions when <span><math><mi>α</mi><mo>≥</mo><mn>5</mn><mo>/</mo><mn>4</mn></math></span>. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><msubsup><mrow><mi>W</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>p</mi></mrow></msubsup></math></span>, in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints <span><math><mo>(</mo><mn>3</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>α</mi><mo>,</mo><mo>∞</mo><mo>,</mo><mi>p</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mn>2</mn><mi>α</mi><mo>/</mo><mi>γ</mi><mo>+</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>α</mi><mo>,</mo><mi>γ</mi><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>. Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff <span><math><msup><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msup></math></span> measure, where <span><math><msub><mrow><mi>η</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>></mo><mn>0</mn></math></span> is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"190 ","pages":"Article 103602"},"PeriodicalIF":2.1000,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: Beyond the Lions exponent\",\"authors\":\"Yachun Li , Peng Qu , Zirong Zeng , Deng Zhang\",\"doi\":\"10.1016/j.matpur.2024.103602\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent <em>α</em> can be larger than the Lions exponent 5/4. It is well-known that, due to Lions <span><span>[1]</span></span>, for any <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> divergence-free initial data, there exist unique smooth Leray-Hopf solutions when <span><math><mi>α</mi><mo>≥</mo><mn>5</mn><mo>/</mo><mn>4</mn></math></span>. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><msubsup><mrow><mi>W</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>p</mi></mrow></msubsup></math></span>, in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints <span><math><mo>(</mo><mn>3</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>α</mi><mo>,</mo><mo>∞</mo><mo>,</mo><mi>p</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mn>2</mn><mi>α</mi><mo>/</mo><mi>γ</mi><mo>+</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>α</mi><mo>,</mo><mi>γ</mi><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>. Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff <span><math><msup><mrow><mi>H</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msup></math></span> measure, where <span><math><msub><mrow><mi>η</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>></mo><mn>0</mn></math></span> is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.</p></div>\",\"PeriodicalId\":51071,\"journal\":{\"name\":\"Journal de Mathematiques Pures et Appliquees\",\"volume\":\"190 \",\"pages\":\"Article 103602\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-07-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal de Mathematiques Pures et Appliquees\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021782424001004\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782424001004","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: Beyond the Lions exponent
We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1], for any divergence-free initial data, there exist unique smooth Leray-Hopf solutions when . We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces , in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints and . Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff measure, where is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.
期刊介绍:
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.