Long Time Existence of A Flow of Elliptic Systems

TL;DR

This paper develops a monotone entropy approach to prove the long-term existence of flows for elliptic systems on Riemann surfaces, including Liouville and Toda systems, without topological or coefficient restrictions.

Contribution

It introduces a novel entropy method to establish long-time existence of parabolic flows for broad classes of elliptic systems on Riemann surfaces.

Findings

Proves long-term existence of flows for Liouville and Toda systems.

Does not require topological data of Riemann surfaces.

No positive lower bounds on coefficient functions needed.

Abstract

For elliptic systems defined on Riemann surfaces, Liouville and Toda systems represent two well-known classes exhibiting drastically different solution structures. Over the years, existence results for these systems have highlighted discrepancies due to their unique solution structures. In this work, we aim to construct a monotone entropy form and establish the long-term existence of a flow of parabolic systems. As a result of our main theorem, we can prove existence results for some broad classes of elliptic systems, including both Liouville and Toda systems. The strength of our results is further underscored by the fact that no topological information about the Riemann surfaces is required and no positive lower bound of coefficient functions is postulated.