Stability of the Yang-Mills heat equation for finite action

TL;DR

This paper proves the stability of solutions to the Yang-Mills heat equation with initial data in the critical Sobolev space, using advanced techniques to handle singular coefficients and weak parabolicity.

Contribution

It establishes the stability of solutions for the Yang-Mills heat equation with critical initial data, extending previous existence and uniqueness results.

Findings

Solutions are stable under small perturbations in initial data.

Unique strong solutions exist up to addition of vertical solutions.

Energy inequalities and Neumann domination techniques are effective for a priori estimates.

Abstract

The existence and uniqueness of solutions to the Yang-Mills heat equation over domains in Euclidean three space was proven in a previous paper for initial data lying in the Sobolev space of order one-half, which is the critical Sobolev index for this equation. In the present paper the stability of these solutions will be established. The variational equation, which is only weakly parabolic, and has highly singular coefficients, will be shown to have unique strong solutions up to addition of a vertical solution. Initial data will be taken to be in Sobolev class one-half. The proof relies on an infinitesimal version of the ZDS procedure: one solves first an augmented, strictly parabolic version of the variational equation and then adds to the solution a function which is vertical along the original path. Energy inequalities and Neumann domination techniques will be used to establish apriori initial behavior for solutions.