Sharp well-posedness for the Benjamin-Ono equation

• 成果类型: Article
• 署名作者: Killip, Rowan; Laurens, Thierry; Visan, Monica
• 署名单位: University of California System; University of California Los Angeles; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01250-8
• 发表日期: 2024
• 页码: 999-1054
• 关键词: regularity conservation-laws; inverse scattering transform; smoothing properties; explicit formula; internal waves; cauchy-problem; ill-posedness; schrodinger; EXISTENCE; hierarchy

摘要

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>-(1)|(2). The proof rests on a new gauge transfor-mation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional divi-dends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of G & eacute;rard's explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy