  The existence and uniqueness of solutions to the Yang-Mills heat equation is
proven over three dimensional Euclidean space and over a bounded open convex
set therein. The initial data is taken to lie in the Sobolev space of order one
half, which is the critical Sobolev index for this equation over a three
dimensional manifold. The existence is proven by solving first an augmented,
strictly parabolic equation and then gauge transforming the solution to a
solution of the Yang-Mills heat equation itself. The gauge functions needed to
carry out this procedure lie in the critical gauge group of Sobolev regularity
three halves, which is a complete topological group in a natural metric but is
not a Hilbert Lie group. The nature of this group must be understood in order
to carry out the reconstruction procedure. Solutions to the Yang-Mills heat
equation are shown to be strong solutions modulo these gauge functions. Energy
inequalities and Neumann domination inequalities are used to establish needed
initial behavior properties of solutions to the augmented equation.
