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Global regularity for the Yang–Mills equations on high dimensional Minkowski space
About this Title
Joachim Krieger, Bâtiment des Mathématiques, EPFL, Station 8, CH-1015 Lausanne, Switzerland and Jacob Sterbenz, Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112
Publication: Memoirs of the American Mathematical Society
Publication Year:
2013; Volume 223, Number 1047
ISBNs: 978-0-8218-4489-2 (print); 978-0-8218-9871-0 (online)
DOI: https://doi.org/10.1090/S0065-9266-2012-00566-1
Published electronically: October 4, 2012
Keywords: wave-equation,
Yang-Mills equations,
critical regularity
MSC: Primary 35L70; Secondary 70S15
Table of Contents
Chapters
- 1. Introduction
- 2. Some Gauge-Theoretic Preliminaries
- 3. Reduction to the “Main a-Priori Estimate”
- 4. Some Analytic Preliminaries
- 5. Proof of the Main A-Priori Estimate
- 6. Reduction to Approximate Half-Wave Operators
- 7. Construction of the Half-Wave Operators
- 8. Fixed Time $L^2$ Estimates for the Parametrix
- 9. The Dispersive Estimate
- 10. Decomposable Function Spaces and Some Applications
- 11. Completion of the Proof
Abstract
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.- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
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