arXiv subjects

Uihyeon Jeong

Publications and source records attributed to Uihyeon Jeong.

4 recordsLinked to original sources

arXiv2026

Classification for the $2$-equivariant harmonic map heat flow and the radial energy-critical nonlinear heat equation in dimension $6$

We consider the global dynamics of finite-energy solutions to the $2$-equivariant harmonic map heat flow from $\bbR^2$ to $\bbS^2$ and $\dot H^1$-bounded radial solutions to the energy-critical nonlinear heat equation in dimension $6$. Building upon soliton resolution, we obtain a classification of their multi-bubble dynamics. First, we prove that all such solutions are global. For the harmonic map heat flow, the number and sign of the bubbles are determined by the energy class of the initial data, while for the nonlinear heat equation the bubble signs necessarily alternate. In both models, the largest scale is determined, up to a positive multiplicative constant, by the spatial tail of the initial datum. The remaining scales concentrate according to exponential and iterated-exponential laws determined by the largest scale. Finally, global bubble trees with an arbitrary number of bubbles exist for both models. For the harmonic map heat flow, this follows directly from the classification; for the nonlinear heat equation, we construct alternating bubble trees with any prescribed radial $\dot H^1$ tail, thereby realizing the scale laws arising in the classification.

math.AP↗

arXiv2026

Quantized blow-up dynamics for Calogero--Moser derivative nonlinear Schrödinger equation

We consider the Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS), an $L^2$-critical nonlinear Schrödinger type equation enjoying a number of numerous structures, such as nonlocal nonlinearity, self-duality, pseudo-conformal symmetry, and complete integrability. In this paper, we construct smooth finite-time blow-up solutions to (CM-DNLS) that exhibit a sequence of discrete blow-up rates, so-called \emph{quantized blow-up rates}. Our strategy is a forward construction of the blow-up dynamics based on modulation analysis. Our main novelty is to utilize the \emph{nonlinear adapted derivative} suited to the \textit{Lax pair structure} and to rely on the \emph{hierarchy of conservation laws} inherent in this structure to control higher-order energies. This approach replaces a repulsivity-based energy method in the bootstrap argument, which significantly simplifies the analysis compared to earlier works. Our result highlights that the integrable structure remains a powerful tool, even in the presence of blow-up solutions. In (CM-DNLS), one of the distinctive features is \emph{chirality}. However, our constructed solutions are not chiral, since we assume the radial (even) symmetry in the gauge transformed equation. This radial assumption simplifies the modulation analysis.

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arXiv2026

Classification of single-bubble blow-up solutions for Calogero--Moser derivative nonlinear Schrödinger equation

We study the Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS), a mass-critical and completely integrable dispersive model. Recent works established finite-time blow-up constructions and soliton resolution, describing the asymptotic behaviors of blow-up solutions. In this paper, we go beyond soliton resolution and provide a sharp classification of finite-time blow-up dynamics in the \textit{single-bubble} regime. Assuming that a solution blows up at time $0<T<\infty$ with a single-soliton profile, we determine all possible blow-up rates. For initial data in $H^{2L+1}(\mathbb{R})$ with $L\ge1$, we prove a dichotomy: either the solution lies in a \emph{quantized regime}, where the scaling parameter satisfies \[ λ(t)\sim (T-t)^{2k},\qquad 1\le k\le L, \] with convergent phase and translation parameters, or it lies in an \emph{exotic regime}, where the blow-up rate satisfies $λ(t)\lesssim (T-t)^{2L+\frac 32}$. To our knowledge, this is the first classification result for quantized blow-up dynamics in the class of dispersive models. We provide a framework for identifying the quantized blow-up rates in classification problems. The proof relies on a modulation analysis combined with the hierarchy of conservation laws provided by the complete integrability of (CM-DNLS). However, it does not use \emph{more refined integrability-based techniques}, such as the inverse scattering method, the method of commuting flows, or the explicit formula. As a result, our analysis applies beyond the chiral solutions.

math.AP↗

arXiv2024

Quantized slow blow-up dynamics for the energy-critical corotational wave maps problem

We study the blow-up dynamics for the energy-critical 1-corotational wave maps problem with 2-sphere target. In arXiv:0911.0692, Raphaël and Rodnianski exhibited a stable finite time blow-up dynamics arising from smooth initial data. In this paper, we exhibit a sequence of new finite-time blow-up rates (quantized rates), which can still arise from well-localized smooth initial data. We closely follow the strategy of the paper arXiv:1301.1859 by Raphaël and Schweyer, who exhibited a similar construction of the quantized blow-up rates for the harmonic map heat flow. The main difficulty in our wave maps setting stems from the lack of dissipation and its critical nature, which we overcome by a systematic identification of correction terms in higher-order energy estimates.

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