Smooth radial quantized Type II blow-up for the three-dimensional energy-critical nonlinear Schrödinger equation
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≥1, smooth radial solutions concentrating a single ground state with the blow-up rate λ(t)=(ℓ+o(1))(T−t)2L for some ℓ>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.
Related
- Threshold Scattering for the Three-Dimensional Focusing Energy-Critical Nonlinear Schrödinger Equation
- Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
- Global well-posedness of the energy-critical nonlinear Schrödinger equation on the three-dimensional torus with additive Lévy noise
- The critical boundary value for a nonlinear Schrödinger equation in a half-space: rigidity and dimensional transition
- Strong and Weak-Type Dispersive Estimates for the Energy-Critical Nonlinear Schrödinger Equation with an Inverse-Square Potential
BibTeX
@misc{kim2026smooth,
title = {{Smooth radial quantized Type II blow-up for the three-dimensional energy-critical nonlinear Schrödinger equation}},
author = {Taegyu Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26127},
eprint = {2610.26127},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26127},
}
RIS
TY - UNPB
TI - Smooth radial quantized Type II blow-up for the three-dimensional energy-critical nonlinear Schrödinger equation
AU - Taegyu Kim
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26127
UR - https://arxiv.org/abs/2610.26127
AB - 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≥1, smooth radial solutions concentrating a single ground state with the blow-up rate λ(t)=(ℓ+o(1))(T−t)^2L for some ℓ>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.
ER -