Weak solution of Parabolic complex Monge-Amp\`ere equation

TL;DR

This paper investigates weak solutions to a parabolic complex Monge-Ampère equation related to the Kähler-Ricci flow, focusing on cases with smooth boundary conditions but irregular initial data.

Contribution

It introduces a framework for analyzing weak solutions of the parabolic complex Monge-Ampère equation with irregular initial conditions.

Findings

Established existence of weak solutions under specified conditions

Connected the equation to the Kähler-Ricci flow dynamics

Provided regularity results for solutions with smooth boundary data

Abstract

We study the equation $\dot{u}=\log\det (u_{\alpha\bar{\beta}})-Au+f(z,t)$ in domains of $\mathbb{C}^n$. This equation has a close connection with the K\"ahler-Ricci flow. In this paper, we consider the case where the boundary condition is smooth and the initial condition is irregular.