arXiv subjects

Bongsuk Kwon

Publications and source records attributed to Bongsuk Kwon.

18 recordsLinked to original sources

arXiv2026

Stable $C^{7/9}$ cusp formation for the Novikov equation

We establish stable cusp formation for the Novikov equation, a cubically nonlinear Camassa--Holm-type equation. We identify an open set of smooth initial data for which the first gradient blow-up produces a cusp with sharp H\"older regularity $C^{7/9}$. This result shows that, in nonlocal wave-breaking problems, the sharp regularity of the cusp is not determined by the nonlocal or nonlinear structure alone. While the conserved $H^1$-type quantity excludes the $C^{1/3}$ cusp associated with Burgers-type gradient blow-up, the precise H\"older exponent is selected by the coupling between the nonlocal term and the algebraic structure of the nonlinearity. In the Novikov equation, this coupling yields the exponent $7/9$, rather than the $3/5$ exponent known for the Camassa--Holm and Hunter--Saxton equations. The main difficulty is that the naive high-frequency limit retains the cubic character of the equation and therefore does not exhibit a self-similar leading flow. We overcome this by introducing a Galilean-type change of variables around a nonzero background, which reveals a quadratic Hunter--Saxton-type leading equation. Its self-similar profiles determine the $C^{7/9}$ cusp, while the nonlocal and cubic remainders are controlled perturbatively in modulated similarity variables.

math.AP↗

arXiv2026

$L^2$-contraction and asymptotic stability of Cahn--Hilliard fronts

We study the stability of transition fronts for the one-dimensional Cahn--Hilliard equation. More precisely, we prove that for any H\"older initial datum sufficiently close to a front in the $L^2$ norm, the corresponding solution of the Cahn--Hilliard equation exists globally and converges, up to a dynamical shift, to the front as time tends to infinity. For the proof, we develop an $L^2$-stability framework adapted to Cahn--Hilliard fronts. The key ingredient is a nontrivial second-order Poincar\'e-type inequality that reveals a coercive structure in the indefinite quadratic energy form associated with the linearized operator about the front, once the dynamical shift is taken into account. This yields an $L^2$-contraction estimate and, via a far-field semigroup argument, asymptotic orbital stability of the front in the unweighted $L^2$ topology. The stability analysis requires only the $L^2$-smallness of the initial perturbation; no higher-order smallness, spatial localization, or moment assumptions are imposed.

math.AP↗

arXiv2026

Asymptotic self-similar blow-up for the regularized Saint-Venant equations

We investigate singularity formation in the regularized Saint--Venant (rSV) equations, a conservative, non-dispersive shallow water system that is formally regarded as a Hamiltonian regularization of the isentropic Euler equations. While it is known that smooth solutions to the rSV system can develop gradient blow-up in finite time, the precise structure of such singularities has not been rigorously characterized. In this work, we establish stability of self-similar blow-up profiles of the Hunter--Saxton equation within the rSV framework, using a nonlinear bootstrap argument in dynamically rescaled coordinates. Our analysis captures the detailed space-time dynamics of solutions near the singularity, and proves their sharp $C^{3/5}$ H\"older regularity at the singular time. This regularity differs from the $C^{1/3}$ H\"older regularity of the cubic-root singularities found in the compressible Euler and inviscid Burgers equations. This contrast highlights the structural influence of the Hamiltonian regularization on singularity formation. To illuminate this effect, we also show that the same $C^{3/5}$ blow-up profile emerges in the regularized Burgers equation, a scalar analogue of the rSV system.

math.AP↗

arXiv2024

Sharp regularity of gradient blow-up solutions in the Camassa-Holm equation

We study the formation of singularities in the Camassa-Holm (CH) equation, providing a detailed description of the blow-up dynamics and identifying the precise H\"older regularity of the gradient blow-up solutions. To this end, we first construct self-similar blow-up profiles and examine their properties, including the asymptotic behavior at infinity, which determines the type of singularity. Using these profiles as a reference and employing modulation theory, we establish global pointwise estimates for the blow-up solutions in self-similar variables, thereby demonstrating the stability of the self-similar profiles we construct. Our results indicate that the solutions, evolving from smooth initial data within a fairly general open set, form $C^{3/5}$ cusps as the first singularity in finite time. These singularities are analogous to \emph{pre-shocks} emerging in the Burgers equation, exhibiting unbounded gradients while the solutions themselves remain bounded. However, the nature of the singularity differs from that of the Burgers equation, which is a cubic-root singularity, i.e., $C^{1/3}$. Our work provides the first proof that generic pre-shocks of the CH equation exhibit $C^{3/5}$ H\"older regularity. It is our construction of new self-similar profiles, incorporating the precise leading-order correction to the CH equation in the blow-up regime, that enables us to identify the sharp H\"older regularity of the singularities and to capture the detailed spatial and temporal dynamics of the blow-up. We also show that the generic singularities developed in the Hunter-Saxton (HS) equation are of the same type as those in the CH equation.

math.AP↗

arXiv2024

Long-Time Behavior towards Shock Profiles for the Navier-Stokes-Poisson System

We study the stability of shock profiles in one spatial dimension for the isothermal Navier-Stokes-Poisson (NSP) system, which describes the dynamics of ions in a collision-dominated plasma. The NSP system admits a one-parameter family of smooth traveling waves, called shock profiles, for a given far-field condition satisfying the Lax entropy condition. In this paper, we prove that if the initial data is sufficiently close to a shock profile in $H^2$-norm, then the global solution of the Cauchy problem tends to the smooth manifold formed by the parametrized shock profiles as time goes to infinity. This is achieved using the method of $a$-contraction with shifts, which does not require the zero mass condition.

math.AP↗

arXiv2024

Singularity formation of hydromagnetic waves in cold plasma

We study $C^1$ blow-up of the compressible fluid model introduced by Gardner and Morikawa, which describes the dynamics of a magnetized cold plasma. We propose sufficient conditions that lead to $C^1$ blow-up. In particular, we find that smooth solutions can break down in finite time even if the gradient of initial velocity is identically zero. The density and the gradient of the velocity become unbounded as time approaches the lifespan of the smooth solution. The Lagrangian formulation reduces the singularity formation problem to finding a zero of the associated second-order ODE.

math.AP↗

arXiv2024

Delta-shock for the pressureless Euler-Poisson system

We study singularity formation for the pressureless Euler-Poisson system of cold ion dynamics. In contrast to the Euler-Poisson system with pressure, when its smooth solutions experience $C^1$ blow-up, the $L^\infty$ norm of the density becomes unbounded, which is often referred to as a delta-shock. We provide a constructive proof of singularity formation to obtain an exact blow-up profile and the detailed asymptotic behavior of the solutions near the blow-up point in both time and space. Our result indicates that at the blow-up time $t=T_\ast$, the density function is unbounded but is locally integrable with the profile of $\rho(x,T_\ast) \sim (x-x_*)^{-2/3}$ near the blow-up point $x=x_\ast$. This profile is not yet a Dirac measure. On the other hand, the velocity function has $C^{1/3}$ regularity at the blow-up point. Loosely following our analysis, we also obtain an exact blow-up profile for the pressureless Euler equations.

math.AP↗

arXiv2024

Structure of singularities for the Euler-Poisson system of ion dynamics

We study the formation of singularity for the isothermal Euler-Poisson system arising from plasma physics. Contrast to the previous studies yielding only limited information on the blow-up solutions, for instance, sufficient conditions for the blow-up and the temporal blow-up rate along the characteristic curve, we rather give a constructive proof of singularity formation from smooth initial data. More specifically, employing the stable blow-up profile of the Burgers equation in the self-similar variables, we establish the global stability estimate in the self-similar time, which yields the asymptotic behavior of blow-up solutions near the singularity point. Our analysis indicates that the smooth solution to the Euler-Poisson system can develop a cusp-type singularity; it exhibits $C^1$ blow-up in a finite time, while it belongs to $C^{1/3}$ at the blow-up time, provided that smooth initial data are sufficiently close to the blow-up profile in some weighted $C^4$-topology. We also present a similar result for the isentropic case, and discuss noteworthy differences in the analysis.

math.AP↗

arXiv2024

Approximate solutions for the Vlasov--Poisson system with boundary layers

We construct the approximate solutions to the Vlasov--Poisson system in a half-space, which arises in the study of the quasi-neutral limit problem in the presence of a sharp boundary layer, referred as to the plasma sheath in the context of plasma physics. The quasi-neutrality is an important characteristic of plasmas and its scale is characterized by a small parameter, called the Debye length. We present the approximate equations obtained by a formal expansion in the parameter and study the properties of the approximate solutions. Moreover, we present numerical experiments demonstrating that the approximate solutions converge to those of the Vlasov--Poisson system as the parameter goes to zero.

math-ph↗

arXiv2022

Formation of Singularities in Plasma Ion Dynamics

We study the formation of singularity for the Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time. We establish criteria for $C^1$ blow-up of the Euler-Poisson system, both for the isothermal and pressureless cases. In particular, our blow-up condition for the presureless model does not require that the gradient of velocity is negatively large. In fact, our result particularly implies that the smooth solutions can break down even if the gradient of initial velocity is trivial. For the isothermal case, we prove that smooth solutions leave $C^1$ class in a finite time when the gradients of the Riemann functions are initially large.

math.AP↗

arXiv2021

Accelerating Optimal Experimental Design for Robust Synchronization of Uncertain Kuramoto Oscillator Model Using Machine Learning

Recent advances in objective-based uncertainty quantification (objective-UQ) have shown that such a goal-driven approach for quantifying model uncertainty is extremely useful in real-world problems that aim at achieving specific objectives based on complex uncertain systems. Central to this objective-UQ is the concept of mean objective cost of uncertainty (MOCU), which provides effective means of quantifying the impact of uncertainty on the operational goals at hand. MOCU is especially useful for optimal experimental design (OED) as the potential efficacy of an experimental (or data acquisition) campaign can be quantified by estimating the MOCU that is expected to remain after the campaign. However, MOCU-based OED tends to be computationally expensive, which limits its practical applicability. In this paper, we propose a novel machine learning (ML) scheme that can significantly accelerate MOCU computation and expedite MOCU-based experimental design. The main idea is to use an ML model to efficiently search for the optimal robust operator under model uncertainty, a necessary step for computing MOCU. We apply the proposed ML-based OED acceleration scheme to design experiments aimed at optimally enhancing the control performance of uncertain Kuramoto oscillator models. Our results show that the proposed scheme results in up to 154-fold speed improvement without any degradation of the OED performance.

math.OC↗

arXiv2021

Optimal Experimental Design for Uncertain Systems Based on Coupled Differential Equations

We consider the optimal experimental design problem for an uncertain Kuramoto model, which consists of N interacting oscillators described by coupled ordinary differential equations. The objective is to design experiments that can effectively reduce the uncertainty present in the coupling strengths between the oscillators, thereby minimizing the cost of robust control of the uncertain Kuramoto model. We demonstrate the importance of quantifying the operational impact of the potential experiments in designing optimal experiments.

math.OC↗

arXiv2020

Linear Stability of solitary waves for the isothermal Euler-Poisson system

We study the asymptotic linear stability of a two-parameter family of solitary waves for the isothermal Euler-Poisson system. When the linearized equations about the solitary waves are considered, the associated eigenvalue problem in $L^2$ space has a zero eigenvalue embedded in the neutral spectrum, i.e., there is no spectral gap. To resolve this issue, use is made of an exponentially weighted $L^2$ norm so that the essential spectrum is strictly shifted into the left-half plane, and this is closely related to the fact that solitary waves exist in the super-ion-sonic regime. Furthermore, in a certain long-wavelength scaling, we show that the Evans function for the Euler-Poisson system converges to that for the Korteweg-de Vries (KdV) equation as an amplitude parameter tends to zero, from which we deduce that the origin is the only eigenvalue on its natural domain with algebraic multiplicity two. We also show that the solitary waves are spectrally stable in $L^2$ space. Moreover, we discuss (in)stability of large amplitude solitary waves.

math.AP↗

arXiv2018

Small amplitude limit of solitary waves for the Euler-Poisson system

The one-dimensional Euler-Poisson system arises in the study of phenomena of plasma such as plasma solitons, plasma sheaths, and double layers. When the system is rescaled by the Gardner-Morikawa transformation, the rescaled system is known to be formally approximated by the Korteweg-de Vries (KdV) equation. In light of this, we show existence of solitary wave solutions of the Euler-Poisson system in the stretched moving frame given by the transformation, and prove that they converge to the solitary wave solution of the associated KdV equation as the small amplitude parameter tends to zero. Our results assert that the formal expansion for the rescaled system is mathematically valid in the presence of solitary waves and justify Sagdeev's formal approximation for the solitary wave solutions of the pressureless Euler-Poisson system. Our work extends to the isothermal case.

math.AP↗

arXiv2017

On the synchronization theory of Kuramoto oscillators under the effect of inertia

We investigate the synchronized collective behavior of the Kuramoto oscillators with inertia effect. Both the frequency synchronization for nonidentical case and the phase synchronization for identical case are in view. As an application of our general theory, we show the unconditional frequency synchronization for the three-nonidentical-oscillator case.

math.DS↗

arXiv2016

The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations

We present a new hydrodynamic model consisting of the pressureless Euler equations and the isentropic compressible Navier-Stokes equations where the coupling of two systems is through the drag force. This coupled system can be derived, in the hydrodynamic limit, from the particle-fluid equations that are frequently used to study the medical sprays, aerosols and sedimentation problems. For the proposed system, we first construct the local-in-time classical solutions in an appropriate $L^2$ Sobolev space. We also establish the \emph{a priori} large-time behavior estimate by constructing a Lyapunov functional measuring the fluctuation of momentum and mass from the averaged quantities, and using this together with the bootstrapping argument, we obtain the global classical solution. The large-time behavior estimate asserts that the velocity functions of the pressureless Euler and the compressible Navier-Stokes equations are aligned exponentially fast as time tends to infinity.

math.AP↗

arXiv2008

Asymptotic Behavior of Multidimensional Scalar Relaxation Shocks

We establish pointwise bounds for the Green function and consequent linearized stability for multidimensional planar relaxation shocks of general relaxation systems whose equilibrium model is scalar, under the necessary assumption of spectral stability. Moreover, we obtain nonlinear $L^{2}$ asymptotic behavior/sharp decay rate of perturbed weak shocks of general simultaneously symmetrizable relaxation systems, under small $L^{1}\cap H^{[d/2]+3}$ perturbations with first moment in the normal direction to the front.

math.AP↗

arXiv2008

Structural Conditions for Full MHD Equations

In this paper, we investigate the characteristic structure of the full equations of magnetohydrodynamics (MHD) and show that it satisfies the hypotheses of a general variable-multiplicity stability frame- work introduced by Metivier and Zumbrun, thereby extending to the general case various results obtained by Metivier and Zumbrun for the isentropic equations of MHD.

math.AP↗