{"title":"线性索波列夫传输中的极端时间间歇性:几乎平滑的非唯一解","authors":"Alexey Cheskidov, Xiaoyutao Luo","doi":"10.2140/apde.2024.17.2161","DOIUrl":null,"url":null,"abstract":"<p>We revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup></math> for all <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\n<mo><</mo>\n<mi>∞</mi></math> in space dimensions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\n<mo>≥</mo> <mn>2</mn></math> whose transport equations admit nonunique weak solutions belonging to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>p</mi></mrow></msubsup><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math> for all <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\n<mo><</mo>\n<mi>∞</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi>\n<mo>∈</mo>\n<mi>ℕ</mi></math>. In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna–Lions theory is essential. The same result also holds for transport-diffusion equations with diffusion operators of arbitrarily large order in any dimensions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\n<mo>≥</mo> <mn>2</mn></math>. </p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Extreme temporal intermittency in the linear Sobolev transport: Almost smooth nonunique solutions\",\"authors\":\"Alexey Cheskidov, Xiaoyutao Luo\",\"doi\":\"10.2140/apde.2024.17.2161\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup></math> for all <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi>\\n<mo><</mo>\\n<mi>∞</mi></math> in space dimensions <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>d</mi>\\n<mo>≥</mo> <mn>2</mn></math> whose transport equations admit nonunique weak solutions belonging to <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>p</mi></mrow></msubsup><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math> for all <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>p</mi>\\n<mo><</mo>\\n<mi>∞</mi></math> and <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>k</mi>\\n<mo>∈</mo>\\n<mi>ℕ</mi></math>. In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna–Lions theory is essential. The same result also holds for transport-diffusion equations with diffusion operators of arbitrarily large order in any dimensions <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>d</mi>\\n<mo>≥</mo> <mn>2</mn></math>. </p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-07-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2140/apde.2024.17.2161\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/apde.2024.17.2161","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Extreme temporal intermittency in the linear Sobolev transport: Almost smooth nonunique solutions
We revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity for all in space dimensions whose transport equations admit nonunique weak solutions belonging to for all and . In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna–Lions theory is essential. The same result also holds for transport-diffusion equations with diffusion operators of arbitrarily large order in any dimensions .
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.