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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

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
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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.

Table of Contents

  • 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
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