Some measure theoretic aspect of HJ equations

We will discuss the commutators of Hamilton-Jacobi equations and discuss the controllability issue. This makes us observe some measure theoretic aspect of HJ equation from mass transport point of view. As an application , I will also discuss the Young measure obtained from the variational vector field given by the intrinsic propagation of singularities. This is based on recent joint work with Piermarco Cannarsa and Jiahui Hong.