Regularizing properties of Complex Monge-Amp\`ere flows

We study the regularizing properties of complex Monge-Amp`ere flows on a K"ahler manifold $(X,\omega )$ when the initial data are $\omega$-psh functions with zero Lelong number at all points. We prove that the general Monge-Amp`ere flow has a solution which is immediately smooth. We also prove the uniqueness and stability of solution.