A general comparison principle for the pluripotential complex Monge-Amp\`ere flow

TL;DR

This paper establishes a comparison principle for pluripotential complex Monge-Ampère flows, ensuring uniqueness of weak solutions and analyzing their long-term behavior under specific conditions.

Contribution

It introduces a new comparison principle for Monge-Ampère flows with specific right-hand sides, leading to uniqueness and long-term analysis of solutions.

Findings

Proved a comparison principle for the Monge-Ampère flow.

Established uniqueness of weak solutions to the Cauchy-Dirichlet problem.

Analyzed the long-term behavior of solutions.

Abstract

We prove a comparison principle for the pluripotential complex Monge-Amp\`ere flows for the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere measure of a bounded plurisubharmonic function. As a consequence, we obtain the uniqueness of the weak solution to the pluripotential Cauchy-Dirichlet problem. We also study the long-term behavior of the solution under some assumption.