Non-simple blow-up for the Chern--Simons--Higgs equation: A priori analysis and constructions
Abstract
Blow-up in the self-dual Chern--Simons--Higgs equation has a rich multiscale structure, but the detailed theory has largely focused on the simple regime, where one entire profile resolves each concentration point. By contrast, non-simple blow-up is analytically more delicate: several interacting regular cores must be resolved simultaneously as they collapse toward a single vortex while remaining separated on their own scales. Because of this difficulty, the existence of such solutions has remained a long-standing open problem. To the best of our knowledge, we prove the first existence result for non-simple blow-up solutions of this equation, covering in particular the previously unresolved finite-height Chern--Simons regime. We also show that non-simple clusters have equal limiting core masses, a common profile type, and regular-polygon arrangements. We determine their possible total masses and derive a mass-gap criterion that rules out such blow-up for certain vortex configurations. Our quantitative profile and moment estimates for general finite-height clusters guide the design of the approximate solution and clarify the core interactions underlying the finite-height construction. A separate construction on the unit disk produces two mean-field cores along a sequence with varying Dirichlet traces.
Related
BibTeX
@misc{lee2026nonsimple,
title = {{Non-simple blow-up for the Chern--Simons--Higgs equation: A priori analysis and constructions}},
author = {Youngae Lee and Lei Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20042},
eprint = {2609.20042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20042},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-simple blow-up for the Chern--Simons--Higgs equation: A priori analysis and constructions
AU - Youngae Lee
AU - Lei Zhang
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20042
UR - https://arxiv.org/abs/2609.20042
AB - Blow-up in the self-dual Chern--Simons--Higgs equation has a rich multiscale structure, but the detailed theory has largely focused on the simple regime, where one entire profile resolves each concentration point. By contrast, non-simple blow-up is analytically more delicate: several interacting regular cores must be resolved simultaneously as they collapse toward a single vortex while remaining separated on their own scales. Because of this difficulty, the existence of such solutions has remained a long-standing open problem. To the best of our knowledge, we prove the first existence result for non-simple blow-up solutions of this equation, covering in particular the previously unresolved finite-height Chern--Simons regime. We also show that non-simple clusters have equal limiting core masses, a common profile type, and regular-polygon arrangements. We determine their possible total masses and derive a mass-gap criterion that rules out such blow-up for certain vortex configurations. Our quantitative profile and moment estimates for general finite-height clusters guide the design of the approximate solution and clarify the core interactions underlying the finite-height construction. A separate construction on the unit disk produces two mean-field cores along a sequence with varying Dirichlet traces.
ER -