Long time behavior of leafwise heat flow for Riemannian foliations
  For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of F. 
Jesus A. Alvarez Lopez and Yuri A. Kordyukov