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

IF 2.1 1区 数学 Q1 MATHEMATICS
Yachun Li , Peng Qu , Zirong Zeng , Deng Zhang
{"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 ,&nbsp;Peng Qu ,&nbsp;Zirong Zeng ,&nbsp;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>&gt;</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 ,&nbsp;Peng Qu ,&nbsp;Zirong Zeng ,&nbsp;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>&gt;</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}
引用次数: 0

摘要

我们研究了三维环上的超发散纳维-斯托克斯方程,其中粘度指数可以大于 Lions 5/4 指数。众所周知,由于 Lions 的存在,对于发散为零的任何初始数据,当......时存在唯一的正则 Leray-Hopf 解。我们证明,即使在这种高耗散机制下,考虑到 Ladyženskaja-Prodi-Serrin 准则,唯一性在超临界空间中也是失效的。非唯一性在强意义上得到了证明,特别是在端点和......处的最优性。此外,所构建的解与初始时间邻域内唯一的 Leray-Hopf 解重合,更微妙的是,在 Hausdorff 量为零的奇异时间分形集(其中是一个给定的小数)外允许部分正则性。这些结果还提供了超临界 Lebesgue 和 Besov 空间中的非唯一性最优性。此外,我们还证明了超耗散 Navier-Stokes 方程的强零粘性极限结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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 L2 divergence-free initial data, there exist unique smooth Leray-Hopf solutions when α5/4. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces LtγWxs,p, 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 (3/p+12α,,p) and (2α/γ+12α,γ,). 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 Hη measure, where η>0 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.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
4.30
自引率
0.00%
发文量
84
审稿时长
6 months
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信