{
  "abstract": "We consider the focusing energy-critical nonlinear Schrödinger equation in three dimensions. In \\cite{Schmid2026CVPDE}, Schmid constructed finite-time type II blow-up solutions with a continuum of blow-up rates, while the existence of smooth finite-time type II blow-up solutions remained open. We resolve this problem by constructing, for each integer $L\\ge1$, smooth radial solutions concentrating a single ground state with the blow-up rate $λ(t)=(\\ell+o(1))(T-t)^{2L}$ for some $\\ell>0$. Our proof follows a forward construction based on modulation analysis with the energy method. The main obstacle is the lack of a repulsive structure that can be directly exploited to control the remainder. To overcome this obstacle, we adapt Kato's smoothing theory from asymptotic stability analysis to the forward construction. This yields a global spacetime Morawetz estimate for the linearized flow, providing the spacetime control needed to bound the higher-order energy in place of the monotonicity derived from repulsive structures in previous blow-up constructions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26127",
  "author_total": 1,
  "authors": [
    {
      "name": "Taegyu Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4rt8y14jfshj1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26127",
  "fulltext_url": null,
  "id": "4rt8y14jfshj1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26127",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7223034768",
  "pdf_url": "https://arxiv.org/pdf/2610.26127",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4rt8y14jfshj1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26127",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26127"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Smooth radial quantized Type II blow-up for the three-dimensional energy-critical nonlinear Schrödinger equation",
  "title_text": "Smooth radial quantized Type II blow-up for the three-dimensional energy-critical nonlinear Schrödinger equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/4rt8y14jfshj1",
  "venue": "arXiv",
  "versions": []
}