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Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space
Joachim Krieger, University of Pennsylvania, Philadelphia, PA, and Jacob Sterbenz, University of California, San Diego, La Jolla, CA
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Memoirs of the American Mathematical Society
2013; 99 pp; softcover
Volume: 223
ISBN-10: 0-8218-4489-X
ISBN-13: 978-0-8218-4489-2
List Price: US$69 Individual Members: US$41.40
Institutional Members: US\$55.20
Order Code: MEMO/223/1047

This monograph contains a study of the global Cauchy problem for the Yang-Mills equations on $$(6+1)$$ and higher dimensional Minkowski space, when the initial data sets are small in the critical gauge covariant Sobolev space $$\dot{H}_A^{(n-4)/{2}}$$. Regularity is obtained through a certain "microlocal geometric renormalization" of the equations which is implemented via a family of approximate null Crönstrom gauge transformations. The argument is then reduced to controlling some degenerate elliptic equations in high index and non-isotropic $$L^p$$ spaces, and also proving some bilinear estimates in specially constructed square-function spaces.

• Introduction
• Some gauge-theoretic preliminaries
• Reduction to the "main a-priori estimate"
• Some analytic preliminaries
• Proof of the main a-priori estimate
• Reduction to approximate half-wave operators
• Construction of the half-wave operators
• Fixed time $$L^2$$ estimates for the parametrix
• The dispersive estimate
• Decomposable function spaces and some applications
• Completion of the proof
• Bibliography