  Long time existence and uniqueness of solutions to the Yang-Mills heat
equation is proven over a compact 3-manifold with smooth boundary. The initial
data is taken to be a Lie algebra valued connection form in the Sobolev space
$H_1$. Three kinds of boundary conditions are explored, Dirichlet type, Neumann
type and Marini boundary conditions. The last is a nonlinear boundary
condition, specified by setting the normal component of the curvature to zero
on the boundary. The Yang-Mills heat equation is a weakly parabolic non-linear
equation. We use a technique of Donaldson and Sadun to convert it to a
parabolic equation and then gauge transform the solution of the parabolic
equation back to a solution of the original equation. Apriori estimates are
developed by first establishing a gauge invariant version of the
Gaffney-Friedrichs inequality. A gauge invariant regularization procedure for
solutions is also established.
  Uniqueness holds upon imposition of boundary conditions on only two of the
three components of the connection form because of weak parabolicity. This work
is motivated by possible applications to quantum field theory.
