Existence of non-simple blow-up solutions for the self-dual Chern–Simons–Higgs equation
Does there exist a sequence of solutions to the self-dual Chern–Simons–Higgs equation exhibiting genuinely non-simple blow-up at a single concentration point; that is, can at least two regular cores collapse toward one vortex while remaining mutually separated on their own blow-up scales, rather than being described by a single entire profile?
References
Primary source
Additional references
- Non-simple blow-up for the Chern--Simons--Higgs equation: A priori analysis and constructions — arXiv — Youngae Lee, Lei Zhang
Progress summary
A new unrefereed preprint claims to construct the non-simple blow-up solutions whose existence had remained open.
The problem asks whether genuinely non-simple blow-up solutions exist for the self-dual Chern–Simons–Higgs equation, beyond the previously understood simple regime. The latest source claims a construction rather than merely a conditional analysis.
Known results
A 2023 study analyzed blow-up at vortex points: singular bubbles force simple blow-up, while nonsingular bubbles can produce a multi-bubble alternative; it did not construct the requested solutions. It also established simple blow-up in several low-multiplicity cases, including some with total multiplicity .
September 2026 claimed construction
Youngae Lee and Lei Zhang's preprint reports multi-core clusters, their masses and polygonal structure, together with a separate two-core construction on the disk. If correct, this settles the existence question affirmatively, but the claim is unrefereed and remains unverified.
Current status (as of September 2026): A preprint claims to establish existence through multi-core and two-core constructions, but the result has not been independently verified; absent that claim, the existence problem was open.
Solutions 0
No solutions have been posted yet.