  In this monograph, we develop results on global existence and convergence of
solutions to abstract gradient flows on Banach spaces for a potential function
that obeys the Lojasiewicz-Simon gradient inequality. We prove a
Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over
closed, smooth Riemannian manifolds of arbitrary dimension and apply the
resulting framework to prove new results for the gradient flow equation for the
Yang-Mills energy functional on a principal bundle, with compact Lie structure
group, over a closed, smooth Riemannian manifolds, including the following. If
the initial connection is close enough to a local minimum of the Yang-Mills
energy functional, in a norm sense when the base manifold has arbitrary
dimension or in an energy sense when the base manifold has dimension four, then
the Yang-Mills gradient flow exists for all time and converges to a Yang-Mills
connection. If the initial connection is allowed to have arbitrary energy but
we restrict to the setting of a Hermitian vector bundle over a compact,
complex, Hermitian (but not necessarily Kaehler) surface and the initial
connection has curvature of type (1,1), then the Yang-Mills gradient flow
exists for all time, though bubble singularities may (and in certain cases
must) occur in the limit as time tends to infinity.
