{"title":"本杰明-奥诺方程的夏普好拟性","authors":"Rowan Killip, Thierry Laurens, Monica Vişan","doi":"10.1007/s00222-024-01250-8","DOIUrl":null,"url":null,"abstract":"<p>The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces <span>\\(H^{s}\\)</span> for <span>\\(s>-\\tfrac{1}{2}\\)</span>. The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard’s explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"63 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sharp well-posedness for the Benjamin–Ono equation\",\"authors\":\"Rowan Killip, Thierry Laurens, Monica Vişan\",\"doi\":\"10.1007/s00222-024-01250-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces <span>\\\\(H^{s}\\\\)</span> for <span>\\\\(s>-\\\\tfrac{1}{2}\\\\)</span>. The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard’s explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.</p>\",\"PeriodicalId\":14429,\"journal\":{\"name\":\"Inventiones mathematicae\",\"volume\":\"63 1\",\"pages\":\"\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2024-03-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Inventiones mathematicae\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00222-024-01250-8\",\"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":"Inventiones mathematicae","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00222-024-01250-8","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Sharp well-posedness for the Benjamin–Ono equation
The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces \(H^{s}\) for \(s>-\tfrac{1}{2}\). The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard’s explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.
期刊介绍:
This journal is published at frequent intervals to bring out new contributions to mathematics. It is a policy of the journal to publish papers within four months of acceptance. Once a paper is accepted it goes immediately into production and no changes can be made by the author(s).