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The threshold theorem for the $ (4+1)$-dimensional Yang-Mills equation: An overview of the proof

Authors: Sung-Jin Oh and Daniel Tataru
Journal: Bull. Amer. Math. Soc. 56 (2019), 171-210
MSC (2010): Primary 35L70, 70S15
Published electronically: August 30, 2018
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Abstract: This article is devoted to the energy critical hyperbolic Yang-Mills equation in the $ (4+1)$-dimensional Minkowski space, which is considered by the authors in a sequence of four papers. The final outcome of these papers is twofold: (i) the Threshold Theorem, which asserts that global well-posedness and scattering hold for all topologically trivial initial data with energy below twice the ground state energy; and (ii) the Dichotomy Theorem, which for larger data in arbitrary topological classes provides a choice of two outcomes, either a global scattering solution or a soliton bubbling off. In the last case, the bubbling-off phenomena can happen in one of two ways: (a) in finite time, triggering a finite time blowup; or (b) in infinite time. Our goal here is to first describe the equation and the results, and then to provide an overview of the flow of ideas within their proofs in the above-mentioned four papers.

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Additional Information

Sung-Jin Oh
Affiliation: Korea Institute for Advanced Study, Seoul, Korea 02455

Daniel Tataru
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California, 94720

Received by editor(s): May 22, 2018
Published electronically: August 30, 2018
Additional Notes: The first author was supported by the Miller Research Fellowship from the Miller Institute, UC Berkeley, and the TJ Park Science Fellowship from the POSCO TJ Park Foundation.
The second author was partially supported by the NSF grant DMS-1266182 as well as by a Simons Investigator grant from the Simons Foundation.
Part of the work described here was carried out during the semester-long program “New Challenges in PDE” held at MSRI in the fall of 2015.
Article copyright: © Copyright 2018 American Mathematical Society