Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A flow approach to the Toda system

Yong Luo, Linlin Sun, Guofang Wang

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.

A flow approach to the Toda system

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.
Paper Structure (14 sections, 25 theorems, 411 equations)

This paper contains 14 sections, 25 theorems, 411 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Main results
  4. Strategy of proof of \ref{['main:thm2']}.
  5. Notations
  6. A priori estimates
  7. Global existence
  8. Global convergence
  9. Decay estimates
  10. Partial H1-estimates
  11. Blowup analysis
  12. The lower bound of the functional
  13. Test functions
  14. A Classification of Toda system

Key Result

theorem 1.1

Assume that $A\leq A_N$ and $\rho_i\in(0,4\pi]$ for each $i$. Then for every smooth initial data $u_0=\left(u_{1,0},\dots,u_{N,0}\right)$ satisfying the regularity condition eq:initial-regularity, the flow of Liouville system eq:liouville admits a unique smooth solution defined for all $[0,\infty)$.

Theorems & Definitions (52)

  • theorem 1.1
  • Remark 1.2
  • theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1: Monotonicity formula
  • proof
  • Lemma 2.2: Mass preservation
  • proof
  • Lemma 2.3: Upper bound estimate
  • ...and 42 more