First of all, I am sorry for such naivety. I have not all the intuition in hard analysis as I wish.
I am studying Perelman's work and his big first contribution is to prove that the Ricci flow is in fact a gradient flow. My question is: Why is interesting from the point of view of PDE to know that some time dependent PDE comes from an variational problem? In particular, why is interesting to know the properties of the functional formulation? Why in general is good to search for monotonicity formulas?
I am sorry for being imprecise, but I think these questions are honest.