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Structure of bubbling solutions of Liouville systems with negative singular sources

Yi Gu, Lei Zhang

TL;DR

This work analyzes a Liouville system with negative singular sources on a Riemann surface, focusing on bubbling behavior as parameters cross critical hyper-surfaces. It develops a complete bubble-interaction theory, deriving exact bubbling heights, local masses, and blowup locations, along with leading-term asymptotics for the energy balance when parameters approach critical sets. A central tool is a precise local-global approximation: near each blowup point the solution is close to a radial global Liouville profile, with robust first- and second-order estimates and a careful angular-mode analysis. These results yield tight mass balance constraints, leading-term descriptions, and stringent conditions on the coefficient functions around singular sources, enabling both construction and classification of bubbling solutions in this singular Liouville framework.

Abstract

Liouville systems on Riemann surfaces are instrumental in modeling species growth and particle dynamics in biology and physics. Previously, we established a priori estimates for parameters across regions defined by critical hyper-surfaces. Here, we extend this by giving a priori estimates when parameters are critically positioned. This involves thoroughly characterizing bubble interaction, a key challenge in Liouville systems. During blowup events, we ascertain the exact heights of bubbling solutions about each blowup point, the integrals of each component, and the blowup points' positions. Moreover, as the parameter $ρ$ approaches a critical hyper-surface, we identify a pivotal leading term vital for numerous applications.

Structure of bubbling solutions of Liouville systems with negative singular sources

TL;DR

This work analyzes a Liouville system with negative singular sources on a Riemann surface, focusing on bubbling behavior as parameters cross critical hyper-surfaces. It develops a complete bubble-interaction theory, deriving exact bubbling heights, local masses, and blowup locations, along with leading-term asymptotics for the energy balance when parameters approach critical sets. A central tool is a precise local-global approximation: near each blowup point the solution is close to a radial global Liouville profile, with robust first- and second-order estimates and a careful angular-mode analysis. These results yield tight mass balance constraints, leading-term descriptions, and stringent conditions on the coefficient functions around singular sources, enabling both construction and classification of bubbling solutions in this singular Liouville framework.

Abstract

Liouville systems on Riemann surfaces are instrumental in modeling species growth and particle dynamics in biology and physics. Previously, we established a priori estimates for parameters across regions defined by critical hyper-surfaces. Here, we extend this by giving a priori estimates when parameters are critically positioned. This involves thoroughly characterizing bubble interaction, a key challenge in Liouville systems. During blowup events, we ascertain the exact heights of bubbling solutions about each blowup point, the integrals of each component, and the blowup points' positions. Moreover, as the parameter approaches a critical hyper-surface, we identify a pivotal leading term vital for numerous applications.
Paper Structure (9 sections, 17 theorems, 311 equations, 1 figure)

This paper contains 9 sections, 17 theorems, 311 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results on bubble interactions
  3. Proof of Theorem \ref{['main-thm-cur']}
  4. The profile of bubbling solutions around a singular source
  5. The comparison of bubbling profiles
  6. The leading terms in approximations
  7. Local approximation of bubbling solutions
  8. First order estimates
  9. Second order estimates

Key Result

Theorem 1.1

Let $p_1,...p_{L_1},p_{L_1+1},..,p_{L_2}$ satisfy (reg-p) and $D^2f$ be non-degenerate at critical points of $f$. Then there exists constants $c_t^*>0$ such that if $u^k$ is uniformly bounded as $\rho^k$ tends to $\Gamma_L$ from above non-tangentially, where $u^k=(u_1^k,...,u_n^k)$ is a sequence of solutions to (b-u-sol), $c_t^*$ depends only on $\mathfrak{m}$, the distance from $p_t$ to the near

Figures (1)

  • Figure 1: Example of profile of bubbling solutions

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.3
  • Theorem 2.5
  • Theorem 2.6
  • ...and 16 more