Electronic ISBN:  9780821898710 
Product Code:  MEMO/223/1047.E 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $41.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 223; 2013; 99 ppMSC: Primary 35; Secondary 70;
This monograph contains a study of the global Cauchy problem for the YangMills equations on \((6+1)\) and higher dimensional Minkowski space, when the initial data sets are small in the critical gauge covariant Sobolev space \(\dot{H}_A^{(n4)/{2}}\). 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 nonisotropic \(L^p\) spaces, and also proving some bilinear estimates in specially constructed squarefunction spaces.

Table of Contents

Chapters

1. Introduction

2. Some GaugeTheoretic Preliminaries

3. Reduction to the “Main aPriori Estimate”

4. Some Analytic Preliminaries

5. Proof of the Main APriori Estimate

6. Reduction to Approximate HalfWave Operators

7. Construction of the HalfWave 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


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This monograph contains a study of the global Cauchy problem for the YangMills equations on \((6+1)\) and higher dimensional Minkowski space, when the initial data sets are small in the critical gauge covariant Sobolev space \(\dot{H}_A^{(n4)/{2}}\). 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 nonisotropic \(L^p\) spaces, and also proving some bilinear estimates in specially constructed squarefunction spaces.

Chapters

1. Introduction

2. Some GaugeTheoretic Preliminaries

3. Reduction to the “Main aPriori Estimate”

4. Some Analytic Preliminaries

5. Proof of the Main APriori Estimate

6. Reduction to Approximate HalfWave Operators

7. Construction of the HalfWave 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