Global existence and convergence of solutions to gradient systems and   applications to Yang-Mills gradient flow
  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. 
Paul M. N. Feehan
math.DG math-ph math.AP math.MP