Random data Cauchy theory for supercritical wave equations I: local theory

• 成果类型: Article
• 署名作者: Burq, Nicolas; Tzvetkov, Nikolay
• 署名单位: Universite Paris Saclay; Institut Universitaire de France; Universite de Lille
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-008-0124-z
• 发表日期: 2008
• 页码: 449-475
• 关键词: nonlinear schrodinger-equation; invariant-measures; global existence

摘要

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H-s(M), s < 1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in theHadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in H-s(M), where s >= 1/4 in the case of a boundary less manifold and s >= 8/21 in the case of a manifold with boundary.