Young measures and heat flow problems

Let $M$  and  $N$ be  compact  smooth Riemannian  manifolds without boundaries. Then, for a map  $u:M\to N$ we consider a class ofenergies which includes the the  popular Dirichlet energy and the moregeneral  $p$-energy.   Geometric  or  physical questions  motivate  toinvestigate the critical points of such an energy or the correspondingheat flow problem.  In the case of the Dirichlet energy, the heat flowproblem has  been intensively studied  and is well understood  by now.However, it has turned out  that the case of the $p$-energy ($p\neq2$)is much  more difficult  in many  respects.  We give  a survey  of theknown results for the $p$-harmonic flow and indicate how these resultscan  be extended  to a  larger class  of energy  types by  using Youngmeasure techniques  which have been developed  for stationary problemsin recent years.