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- Publisher
- Springer Journals
- Copyright
- Copyright © 2012 by Springer-Verlag
- Subject
- Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Statistical Physics, Dynamical Systems and Complexity; Classical and Quantum Gravitation, Relativity Theory
- ISSN
- 0010-3616
- eISSN
- 1432-0916
- DOI
- 10.1007/s00220-012-1558-0
- Publisher site
- See Article on Publisher Site
Abstract
Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space H 1. Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Marini boundary conditions. The last is a nonlinear boundary condition, specified by setting the normal component of the curvature to zero on the boundary. The Yang-Mills heat equation is a weakly parabolic nonlinear equation. We use gauge symmetry breaking to convert it to a parabolic equation and then gauge transform the solution of the parabolic equation back to a solution of the original equation. Apriori estimates are developed by first establishing a gauge invariant version of the Gaffney-Friedrichs inequality. A gauge invariant regularization procedure for solutions is also established. Uniqueness holds upon imposition of boundary conditions on only two of the three components of the connection form because of weak parabolicity. This work is motivated by possible applications to quantum field theory.
Journal
Communications in Mathematical Physics – Springer Journals
Published: Sep 1, 2012