We develop new results on global existence and convergence
of solutions to the gradient flow equation for the Yang-Mills energy
functional on a principal bundle, with compact Lie structure group, over
a closed, four-dimensional, Riemannian, smooth manifold, including the
following. If the initial connection is close enough to a minimum of the
Yang-Mills energy functional, in a norm or energy sense, 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. The Lojasiewicz-Simon gradient inequality plays a crucial role in our approach and we develop two versions of that inequality for the
Yang-Mills energy functional.