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The Yang-Mills heat equation with finite action in three dimensions
About this Title
Leonard Gross
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 275, Number 1349
ISBNs: 978-1-4704-5053-3 (print); 978-1-4704-7015-9 (online)
DOI: https://doi.org/10.1090/memo/1349
Published electronically: December 8, 2021
Keywords: Yang-Mills,
heat equation,
weakly parabolic,
gauge groups,
Gaffney-Friedrichs inequality,
Neumann domination
Table of Contents
Chapters
- 1. Introduction
- 2. Statement of results
- 3. Solutions for the augmented Yang-Mills heat equation
- 4. Initial behavior of solutions to the augmented equation
- 5. Gauge groups
- 6. The conversion group
- 7. Recovery of $A$ from $C$
Abstract
The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over $\mathbb {R}^3$ and over a bounded open convex set in $\mathbb {R}^3$. The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.- Marc Arnaudon, Robert O. Bauer, and Anton Thalmaier, A probabilistic approach to the Yang-Mills heat equation, J. Math. Pures Appl. (9) 81 (2002), no. 2, 143–166. MR 1994607, DOI 10.1016/S0021-7824(02)01254-0
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