The regularity of weak solutions of an elliptic complex Monge-Ampere equation is studied on compact Hermitian manifolds. Using the smoothing property for the corresponding parabolic flow, a weak solution is proved to be smooth if the background Hermitian metric satisfies a compatibility condition. The Chern-Ricci flow is an evolution equation of Hermitian metrics on a complex manifold by their Chern-Ricci form. The existence and uniqueness for the Chern-Ricci flow with rough initial data is obtained on compact Hermitian manifolds satisfying a mild assumption. Then we prove the existence of weak solutions of the Chern-Ricci flow through blow downs of exceptional curves, as well as smooth convergence on compact subsets away from image points of the exceptional curves.