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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)
Published electronically: October 4, 2012
Keywords:wave-equation, Yang-Mills equations, critical regularity
MSC: Primary 35L70; Secondary 70S15

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Some gauge-theoretic preliminaries
  • Chapter 3. Reduction to the “main a-priori estimate”
  • Chapter 4. Some analytic preliminaries
  • Chapter 5. Proof of the main a-priori estimate
  • Chapter 6. Reduction to approximate half-wave operators
  • Chapter 7. Construction of the half-wave operators
  • Chapter 8. Fixed time $L^2$ estimates for the parametrix
  • Chapter 9. The dispersive estimate
  • Chapter 10. Decomposable function spaces and some applications
  • Chapter 11. Completion of the proof


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.

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