The threshold conjecture for the energy critical hyperbolic Yang Mills equation

• 成果类型: Article
• 署名作者: Oh, Sung-Jin; Tataru, Daniel
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2021.194.2.1
• 发表日期: 2021
• 页码: 393-473
• 关键词: global well-posedness; klein-gordon equation; wave maps; blow-up; local existence; SCATTERING; REGULARITY; CONSTRUCTION; DYNAMICS; THEOREM

摘要

This article represents the fourth and final part of a four-paper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical 4 + 1-dimensional hyperbolic Yang Mills equation. The Threshold Theorem asserts that topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary topological class with large energy, and provides two exclusive alternatives: Either the solution is global and scatters, or it bubbles off a soliton in either finite time or infinite time. Using the caloric gauge developed in the first paper, the continuation/scattering criteria established in the second paper, and the large data analysis in an arbitrary topological class at optimal regularity in the third paper, here we perform a blow-up analysis that shows that the failure of global well-posedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence of a non-trivial self-similar solution.