Initial behavior of solutions to the Yang-Mills heat equation

TL;DR

This paper investigates the initial behavior of solutions to the Yang-Mills heat equation with rough initial data, focusing on how solution norms behave as time approaches zero.

Contribution

It provides new insights into the small-time behavior of solutions with low-regularity initial data in bounded convex regions and all of 3, extending understanding of the Yang-Mills heat flow.

Findings

Determines the small-time asymptotics of $L^p$ norms of derivatives and curvature.

Analyzes behavior for initial data in $H_{1/2}(M)$.

Provides results for both bounded convex regions and 3.

Abstract

We explore the small-time behavior of solutions to the Yang-Mills heat equation with rough initial data. We consider solutions $A(t)$ with initial value $A_0\in H_{1/2}(M)$, where $M$ is a bounded convex region in $\mathbb{R}^3$ or all of $\mathbb{R}^3$. The behavior, as $t\downarrow 0$, of the $L^p(M)$ norms of the time derivatives of $A(t)$ and its curvature $B(t)$ will be determined for $p=2$ and $6$, along with the $H_1(M)$ norm of these derivatives.