Weak Solutions to the complex Monge-Amp\`ere flows on compact K\"ahler manifolds : general measures on the right-hand side

Abstract

We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Amp\`ere flow on a compact K\"ahler manifold, with the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere measure of a H\"older continuous quasi-plurisubharmonic function. We also prove that for a given semi-positive big from $\theta$, the $t$-slice of the solution is locally H\"older continuous on $\rm{Amp(\theta)}$ for all $t \in (0, T)$. Next, we prove a comparison principle when $d\mu$ is dominated by a Monge-Amp\`ere measure of a bounded quasi-plurisubharmonic function, which implies the uniqueness of the solution.