The heat flow of harmonic maps from a smooth, compact Riemannian manifold without boundary, (M,g) into another smooth, compact Riemannian manifold without boundary (N,h) was first studied in the seminal work of Eells and Sampson when the target manifold (N,h) has non-positive curvature. R.Hamilton studied the case when M has a compact  smooth boundary (and some special cases when N has also a smooth, compact boundary).  Gromov-Schoen studied harmonic maps from M  into a singular Cat(0) space which was used to  understand the p-adic superrigidity of lattices in groups of rank one.  

A key analytic property of such harmonic maps is the Lipschitz continuity, and from which one derives similar Bochner type estimates and vanishing theorems.  As for Eells-Sampson theorem, it is rather natural to study the associated gradient (heat) flow,  and it has been a long open problem for the existence of suitable weak solutions.  In this talk, I shall describe an elliptic approach (which goes back to De Giorgi and also T. Ilmanen in the 1990s) to this problem and to show the global existence of spatially Lipschitz and Holder-1/2 in time solutions. This is a joint work with A.Sagatti, Y.Sire and C.Y. Wang.