arXiv subjects

Jeongheon Park

Publications and source records attributed to Jeongheon Park.

4 recordsLinked to original sources

arXiv2026

Blow-up construction and instability for mass-critical half-wave equation with slightly superthreshold mass

We study the blow-up dynamics for the $L^2$-critical focusing half-wave equation on the real line, a nonlocal dispersive PDE arising in various physical models. As in other mass-critical models, the ground state solution becomes a threshold between the global well-posedness and the existence of a blow-up. The first blow-up construction is due to Krieger, Lenzmann and Raphaël, in which they constructed the minimal mass blow-up solution at the threshold mass. In this paper, we construct finite-time blow-up solutions with mass slightly exceeding the threshold. This is inspired by similar results in the mass-critical NLS by Bourgain and Wang, and their instability by Merle, Raphaël and Szeftel. We exhibit a blow-up profile driven by the rescaled ground state, with a decoupled dispersive radiation component. We rigorously describe the asymptotic behavior of such solutions near the blow-up time, including sharp modulation dynamics. Furthermore, we demonstrate the instability of these solutions by constructing non-blow-up solutions that are arbitrarily close to the blow-up solutions. The main contribution of this work is to overcome the nonlocal setting of half-wave and to extend insights from the mass-critical NLS to a setting lacking pseudo-conformal symmetry.

math.AP↗

arXiv2026

A log-log upper bound on blow-up rates for the mass-critical half-wave equation

We study finite-time blow-up for the one-dimensional focusing mass-critical half-wave equation \begin{equation*} i\partial_tu=|D|u-|u|^2u. \end{equation*} For even initial data with negative energy and mass slightly above the ground-state mass, we prove the log-log upper bound \begin{equation*} \|u(t)\|_{\dot H^{1/2}}\lesssim \left(\frac{\log|\log(T-t)|}{T-t}\right)^{1/2} \quad \text{as}\quad t\uparrow T. \end{equation*} This gives, for the half-wave equation, the same log-log law upper bound as in the mass-critical nonlinear Schrödinger equation. The proof follows a similar strategy developed by Merle and Raphaël, but requires a new construction of the blow-up profile. Main difficulty arises from the nonlocal operator $|D|$ and the absence of pseudo-conformal symmetry. We construct an almost self-similar profile with exponentially small error by combining tail computations carried out to arbitrary order, depending on a dynamical parameter, with Borel integral summation in $Λ$-analytic spaces. Then, in the modulation analysis, we use a local-virial spectral property proved in the companion paper \cite{Park2026arXiv}.

math.AP↗

arXiv2026

Finite-time blow-up for the mass-critical half-wave equation with negative energy

We study the one-dimensional focusing mass-critical half-wave equation \[ i\partial_tu=|D|u-|u|^2u. \] For even initial data with negative energy and mass slightly above the ground-state mass, we prove finite-time blow-up and obtain the upper bound \[ \|u(t)\|_{\dot H^{1/2}} \lesssim \frac{|\log(T-t)|^{1/4}}{\sqrt{T-t}} \qquad\text{as }t\uparrow T. \] This is the first finite-time blow-up result for negative-energy solutions to the mass-critical half-wave equation in the near-ground-state regime. The proof uses the modulation analysis developed by Merle--Raphaël \cite{MerleRaphael2005AnnMath}. For the half-wave equation, the key missing ingredient is a coercivity estimate for a nonlocal quadratic form generated by the scaling direction. Unlike the local NLS, the half-wave equation does not admit the ODE methods used to establish the corresponding coercivity estimate. Instead, we prove the coercivity estimate by an analytic reduction followed by a rigorous computer-assisted proof. The spectral analysis part reduces the coercivity problem to a finite collection of spectral inequalities by combining constrained Morse index arguments with the Birman--Schwinger principle. The computer-assisted part certifies these inequalities by interval arithmetic using a validated approximation of the ground state obtained by compactifying the real line. Together with the modulation analysis, this coercivity theorem gives the finite-time blow-up result.

math.AP↗

arXiv2025

Melting and freezing rates of the radial interior Stefan problem in two dimension

We consider the interior Stefan problem under radial symmetry in two dimension. A water ball surrounded by ice undergoes melting or freezing. We construct a discrete family of global-in-time solutions, both melting and freezing scenarios. The evolution of the free boundary, represented by the radius of the water ball, $λ(t)$ exhibits exponential convergence to a limiting radius value $λ_\infty > 0$, characterized by the asymptotic expression \[ λ(t) = λ_\infty + (1 - λ_\infty)\, e^{-\frac{λ_k}{λ_\infty^2} t + o_{t \to \infty}(1)}, \] where $λ_k$ stands for the $k$-th Dirichlet eigenvalue of the Laplacian on the unit disk for any $k\in \mathbb{N}$. Our approach draws inspiration from the research conducted by Hadžić and Raphaël [24] concerning the exterior radial Stefan problem, which involves an ice ball is surrounded by water. In contrast, the bounded geometry in our setting leads to scenario results in a non-degenerate spectrum, leading to distinctly different long-term behavior. These solutions for each $k$ remain stable under perturbations of co-dimension $k - 1$.

math.AP↗