A flow approach to the Toda system

TL;DR

This paper introduces a flow method for analyzing the Toda system and Liouville systems, providing conditions for convergence, characterizing singularities, and extending previous results to more general, sign-changing cases.

Contribution

It develops a new flow approach for Toda and Liouville systems, including criteria for convergence and blow-up analysis, extending prior work to systems with sign-changing prescribed functions.

Findings

Characterization of finite-time singularities.

Necessary and sufficient conditions for flow convergence.

Extension of existing theorems to systems with sign-changing functions.

Abstract

In this paper we introduce a flow to study the Toda system, which we call {\it Toda flow.} More generally, we introduce a flow of the Liouville systems, formulated as a coupled parabolic system with nonlocal interactions. Finite-time singularities are characterized and both necessary and sufficient conditions for convergence are provided in this general setting, even when the prescribed functions are allowed to change sign. As an application, we prove a global existence for the Toda flow in the critical case without restricting the sign of the prescribed functions. We provide a detailed description of blow-up behavior at infinity and obtain a sharp lower bound for the functional in cases where global convergence fails. By constructing appropriate test functions, we further establish a sufficient condition for the global convergence of the flow. These results are not affected by the sign-changing nature of the prescribed functions, and extend the theorem of Jost, Lin and Wang (Comm. Pure Appl. Math. 59, 526-558, 2006) to systems of multiple equations under this more general and physically relevant condition.