Initial behavior of solutions to the Yang-Mills heat equation

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.