Neumann domination for the Yang-Mills heat equation

TL;DR

This paper improves bounds on solutions to the Yang-Mills heat equation on 3-manifolds using Neumann domination, and shows convergence of gauge-invariant Wilson Loop functions over time.

Contribution

It introduces a Neumann domination technique to obtain sharper pointwise bounds and analyzes the long-time behavior of Wilson Loop functions under various boundary conditions.

Findings

Enhanced pointwise bounds on the magnetic field

Proof of convergence of Wilson Loop functions as time approaches infinity

Applicability to Dirichlet, Neumann, and Marini boundary conditions

Abstract

Long time existence and uniqueness of solutions to the Yang-Mills heat equation have been proven over a compact 3-manifold with boundary for initial data of finite energy. In the present paper we improve on previous estimates by using a Neumann domination technique that allows us to get much better pointwise bounds on the magnetic field. As in the earlier work, we focus on Dirichlet, Neumann and Marini boundary conditions. In addition, we show that the Wilson Loop functions, gauge invariantly regularized, converge as the parabolic time goes to infinity.