arXiv subjects

Seung-Yeal Ha

Publications and source records attributed to Seung-Yeal Ha.

At least 19 recordsLinked to original sources

arXiv2026

Emergent behaviors of the kinetic Motsch-Tadmor model in a phase-spatially extended setting

The kinetic Motsch-Tadmor (in short, KMT) model is a kinetic flocking model with a normalized communication weight. In this paper, we study the emergent dynamics of the phase-spatially extended KMT model. We first establish a global well-posedness theory in the fully noncompact spatial-velocity setting. To this end, we introduce a direct Lagrangian formulation in which the unbounded part of the initial velocity distribution is separated from an interaction-generated bounded remainder. This decomposition allows us to construct global Lagrangian weak solutions without truncating the velocity distribution and to propagate finite phase-space moments. We then investigate the long-time collective behavior of the resulting solutions. When the initial velocity support is compact while the spatial support is allowed to be noncompact, a time-varying effective-region argument yields a uniform contraction mechanism for the normalized interaction and leads to exponential weak support flocking. When both the spatial and velocity supports are noncompact, support-level flocking is in general impossible. Nevertheless, the same effective-region mechanism, combined with the Lagrangian decomposition and a bootstrap argument for the interaction-generated remainder, yields exponential weak moment flocking. In particular, pairwise spatial moments remain uniformly controlled while velocity fluctuations converge exponentially to zero. These results provide a unified framework for the well-posedness and flocking dynamics of the KMT model beyond the compact-support regime.

math.AP↗

arXiv2026

The mean-field limit of the Schrödinger-Lohe model and emergent dynamics

The Schrödinger-Lohe (SL) model is a coupled system of nonlinear Schrödinger equations describing the temporal-spatial evolution of the component wave functions, and it corresponds to the infinite-dimensional counterpart of the Lohe matrix model for quantum synchronization. In this paper, we study a rigorous mean-field limit of the SL model and provide a quantitative estimate on the fluctuations between one-marginal distribution and one particle distribution in 2-Wasserstein distance. For the derived kinetic SL equation, we present sufficient conditions leading to the complete and practical synchronizations.

math.AP↗

arXiv2026

Weak flocking for phase-spatially extended kinetic Cucker-Smale equation in a confining force field

We study the quantitative fast and slow weak flocking of the phase-spatially extended kinetic Cucker-Smale model in confining potential fields. The confining potential is allowed to be nonconvex. In contrast to the phase-spatially confined case, the communication weight may have no positive lower bounds, while spatial and velocity diameters may remain infinite. Moreover, an indefinite Hessian of confining potential prevents the direct use of convexity-based coercive estimates. To overcome these difficulties, we identify three structural assumptions that are sufficient for the noncompact hypocoercive method: quadratic confinement, a globally Lipschitz force, and a strict virial inequality. The admissible class of confining potentials includes genuinely nonconvex radially symmetric ones and localized oscillatory perturbations of the harmonic potential. Weak flocking analysis combines three key ingredients: a microscopic mechanical-energy estimate, a time-varying effective region, and a macroscopic Lyapunov functional. For this, we first establish global existence for initial data with finite second moments. In the exponential mechanical-energy class, we further show the uniqueness and finite-time Osgood-type stability in 1-Wasserstein distance. For polynomially decaying initial mechanical-energy tails, we derive the optimal algebraic decay exponent of the fluctuation energy throughout the admissible communication-decay regime. The optimality is verified by a symmetric countably infinite particle solution for a fixed nonconvex potential. For exponentially decaying tails, a two-stage localization argument yields exponential weak flocking on an optimal exponential time scale. These results show that well-posedness and weak flocking persist even for fully noncompact data under genuinely nonconvex confining forces.

math.AP↗

arXiv2026

Weak stability and random mean-field limit of the phase-spatially extended kinetic Cucker-Smale model

We study the weak stability of the kinetic Cucker-Smale (in short, KCS) model in a phase-spatially extended setting, which can be formally derived from the infinite Cucker-Smale model in the mean-field limit. For a bounded Lipschitz communication weight function, we derive finite-time Osgood-type weak stability for measure-valued solutions with exponential velocity tails and finite spatial second moments. Unlike the phase-spatially confined setting, the solution operator to the KCS model is not Lipschitz continuous with respect to initial data. This is due to the unbounded velocity tail and the corresponding absence of a uniform Lipschitz bound for the alignment force. As an application of weak stability, we obtain an i.i.d. sampling consequence: empirical measures generated from independent initial samples converge to the measure-value solution for the corresponding kinetic model in any finite time interval, in expectation.

math.AP↗

arXiv2026

Uniform-in-time stability and mean-field limit of the Cucker-Smale-type model with noncompact support

We study the uniform-in-time stability and mean-field limit of an infinite Cucker--Smale-type (ICS-type) model in a fully noncompact setting. First, we clarify the main difficulties arising from the original ICS model and its corresponding kinetic Cucker--Smale (KCS) model, when the spatial support is noncompact. In particular, we show that the velocity diameter may remain constant in time, which invalidates the classical approach based on position and velocity diameters. Moreover, we construct a counterexample showing that the original KCS model does not satisfy uniform-in-time stability in the noncompact spatial setting. To overcome these obstacles, we introduce a new ICS-type model whose communication weight depends on a pairwise spatial moment. Under suitable assumptions, we also establish the uniform-in-time stability of the ICS-type model with respect to initial data in the noncompact setting. Finally, we derive the uniform-in-time mean-field limit for the ICS-type model, together with uniform-in-time stability for the corresponding KCS-type model.

math.AP↗

arXiv2026

Particle, kinetic and hydrodynamic models for sea ice floes. Part II: Rotating floes with environmental forces

This paper builds on the multiscale modeling framework introduced in Part I (Deng and Ha, Physica D: Nonlinear Phenomena 483 (2025) 134951) for sea-ice floe dynamics with non-rotating floes to the case with rotational floes, nonlinear contact interactions, Coriolis force and ocean tilt. Building on the particle-kinetic-hydrodynamic hierarchy developed for non-rotating floes, we generalize the particle model to describe ice floes as rigid bodies characterized by position, linear velocity, angular velocity, size, and moment of inertia. The interaction rules now include nonlinear contact forces and torques arising from short-range compression, restitution, and tangential friction laws, together with both oceanic and atmospheric drags that couple translational and rotational motions. These particle descriptions lead to an enriched Vlasov-type kinetic equation posed on an extended phase space, whose moments yield a hydrodynamic system for mass, momentum, and angular-momentum balances. Compared with Part I, the resulting macroscopic equations feature additional hydrodynamic and stress contributions, rotational transport, and dissipative mechanisms stemming from nonlinear collisions. The proposed framework provides a more accurate description of sea-ice floe dynamics and offers a systematic pathway toward multiscale modeling of sea-ice rheology under complex environmental forcing.

math-ph↗

arXiv2026

Uniform-in-time propagation of chaos for Second-Order Consensus-Based Optimization

We study second-order Consensus-Based Optimization (CBO), a derivative-free global optimization algorithm in which the consensus force and the multiplicative exploratory noise act on particle velocities. We prove quantitative uniform-in-time propagation of chaos for the unmodified second-order CBO dynamics, together with an almost uniform-in-time stability estimate for the microscopic particle system. The proof is not a direct adaptation of the first-order CBO argument. Although both first- and second-order CBO have multiplicative noise that degenerates near consensus and a shift-invariant weighted interaction, the kinetic model has an additional structural obstruction: the consensus mechanism and the stochastic forcing act only on the velocity variable, while the position variable evolves by transport. Thus spatial concentration has to be recovered indirectly through velocity dissipation. Moreover, the shift-invariant interaction leaves a translation mode that is not directly damped by the consensus force, so a standard synchronous coupling in the Euclidean phase-space distance does not close uniformly in time. The main idea of the paper is to introduce shifted internal variables that separate the contracting fluctuation modes from the undamped translation mode. In these variables we build a Lyapunov functional with a position-velocity cross term and prove exponential decay of centered moments. This decay is the mechanism that makes the time-dependent coupling coefficient integrable. Combining it with uniform-in-time raw moment bounds, concentration inequalities, stability estimates for the weighted mean, and a Monte Carlo estimate, we obtain the classical Monte Carlo rate for propagation of chaos uniformly in time. The system-to-system stability estimate avoids the sampling error and yields the faster rate $O(J^{-q})$.

math.PR↗

arXiv2026

Emergent behaviors of the singular continuum Kuramoto model and its graph limit

We study the emergent dynamics of the singular continuum Kuramoto model (in short, SCKM) and its graph limit. The SCKM takes the form of an integro-differential equation exhibiting two types of nonlocal singularities: a nonlocal singular interaction weight and a nonlocal singular alignment force. The natural frequency function determines the emergent dynamics of the SCKM, and we emphasize that singularity plays a crucial role in the occurrence of sticking phenomena. For the identical natural frequency function, we derive the complete phase synchronization in finite time under a suitable set of conditions for system parameters and initial data. In contrast, for a nonidentical natural frequency function, we show the emergence of practical synchronization, meaning that the phase diameter is proportional to the inverse of coupling strength asymptotically. Furthermore, we rigorously establish a graph limit from the singular Kuramoto model with a finite system size to the SCKM. We also provide several numerical simulations to illustrate our theoretical results.

math.DS↗

arXiv2026

Emergent behaviors of Winfree oscillators on special orthogonal group

We propose a generalized matrix-valued synchronization model which can be regarded as matrix generalization of the classical Winfree model to the special orthogonal group, and we provide several sufficient frameworks leading to the emergent behaviors of the Winfree matrix model. For $SO(2)$ case, the proposed model reduces to the classical Winfree model. For the general (non-identical) case, we prove the existence of a positively invariant trapping region, establish a leader--follower mechanism in which sufficiently strong coupling draws all oscillators into a neighborhood of the identity whenever at least one oscillator is initially nearby, and show $\ell^1$-exponential stability of solutions, from which we deduce existence, uniqueness, and exponential convergence to an equilibrium. In the identical-oscillator regime, we show that complete state synchronization and oscillator death both occur exponentially fast with an explicit decay rate, and we classify all equilibrium configurations as solutions to a fixed-point equation for the mean influence.

math.DS↗

arXiv2026

On the weak flocking of the kinetic Cucker-Smale model in a fully non-compact support setting

We study the emergent behaviors of the weak solutions to the kinetic Cucker-Smale (in short, KCS) model in a non-compact spatial-velocity support setting. Unlike the compact support situation, non-compact support of a weak solution can cause a communication weight to have zero lower bounds, and position difference does not have a uniformly linear growth bound. These cause the previous approach based on the nonlinear functional approach for spatial and velocity diameters to break down. To overcome these difficulties, we derive refined estimates on the upper bounds for the second-order spatial-velocity moments and show the uniqueness of the weak solution using the estimate on the deviation of particle trajectories. For the estimate of emergent dynamics, we consider two classes of distribution functions with decaying properties (an exponential decay or polynomial decay) in phase space, and then verify that the second moment for the velocity deviation from an average velocity tends to zero asymptotically, while the second moment for spatial deviation from the center of mass remains bounded uniformly in time. This illustrates the robustness of the mono-cluster flocking dynamics of the KCS model even for fully non-compact support settings in phase space and generalizes earlier results on flocking dynamics in a compact support setting.

math.AP↗

arXiv2026

Relaxation dynamics of the continuum Kuramoto model with non-integrable kernels

We study the asymptotic behavior of the continuum Kuramoto model with a fractional Laplacian-type kernel. For this, we construct global weak solutions via a two-parameter regularization procedure using a kernel truncation with fractional dissipation. Using a priori uniform estimates derived in fractional Sobolev spaces, we employ compactness arguments to construct global weak solutions to the singular continuum Kuramoto model. Furthermore, we also establish an exponential relaxation toward the initial phase average in $L^2$-norm under suitable assumptions on initial data and system parameters. These findings provide a rigorous characterization of the existence of solutions and the emergent dynamics of Kuramoto ensembles under physically important strongly singular interactions, including power-law singular kernels and Coulomb-type kernels.

math.AP↗

arXiv2026

Existence of periodic measure-valued solutions to the nonlocal continuity equation via optimal transport

We investigate the existence of periodic solutions for a class of nonlocal continuity equations, which include mean-field equations derived from systems of coupled oscillators. While periodic solutions at the particle level have been studied through the construction of a Poincaré map on a section of an invariant set, extending this analysis to the level of continuity equations presents nontrivial challenges. In particular, setting an appropriate topology for the infinite-dimensional space to show invariance and apply the fixed point argument is not easy. To overcome this difficulty, we use fixed point theorem for geodesically convex spaces constructed by optimal transportation. Specifically, from the disintegration with respect to stationary variable, we define a metric using the Wasserstein-$2$ distance over one-dimensional space, which yields a $CAT(0)$ space. In this topology, we construct an invariance set of probability measures and prove the existence of the periodic measure-valued solution from Schauder's fixed point theorem on geodesically convex spaces. As a corollary, our method directly gives an existence of periodic graph measure solution.

math.DS↗

arXiv2026

CBO algorithm with average drift and applications to portfolio optimization

We propose a consensus based optimization algorithm with average drift (in short Ad-CBO) and provide a theoretical framework for it. In the theoretical analysis, we show that particle solutions to Ad-CBO converge to a global minimizer. In numerical simulations, we examine Ad-CBO's performance in optimizing static and dynamic objective functions. As a real-time application, we test the efficiency of Ad-CBO to find the optimal portfolio given stochastically evolving multi-asset prices in a financial market. The proposed Ad-CBO exhibits higher searching speed, lower tracking errors and regret bound than the CBO without stochastic diffusion

math.OC↗

arXiv2026

On the kinetic equation arising from the large-scale limit of the Cucker-Smale model

We propose a large-scale scaling viewpoint for deriving mesoscopic dynamics from interacting particle systems and apply it to the Cucker--Smale flocking model. In contrast with the classical mean-field regime leading to the Vlasov-type Cucker--Smale equation with spatially nonlocal (convolution) alignment force, our scaling yields a kinetic equation whose alignment field becomes local in space and nonlocal only in velocity. For the spatially homogeneous case, we obtain an explicit solution and derive quantitative flocking rates. For the spatially inhomogeneous equation we establish a local well-posedness in $W^{1,\infty}$ and in $C_b^{1,α}$, highlighting the additional difficulties caused by the absence of a convolution structure. Moreover, for sufficiently small interaction strength we present a global well-posedness and a forward-in-time $L^1$ asymptotic completeness property. Finally, we investigate mono-kinetic solutions and exhibit finite-time blow-up scenarios.

math.AP↗

arXiv2026

Finite-time flocking of an infinite set of Cucker-Smale particles with sublinear velocity couplings

We study finite-time flocking for an infinite set of Cucker-Smale particles with sublinear velocity coupling under fixed and switching sender networks. For this, we use a component-wise diameter framework and exploit sub-linear dissipation mechanisms, and derive sufficient conditions for finite-time flocking equipped with explicit alignment-time estimate. For a fixed sender network, we establish component-wise finite-time flocking results under both integrable and non-integrable communication weights. When communication weight function is non-integrable, finite-time flocking is guaranteed for any bounded initial configuration. We further extend the flocking analysis to switching sender networks and show that finite-time flocking persists under mild assumptions on the cumulative influence of time-varying sender weights. The proposed framework is also applicable to both finite and infinite systems, and it yields alignment-time estimates that do not depend on the number of agents.

math.DS↗

arXiv2025

Emergent behaviors of relativistic thermodynamic flocks with Synge energy

Collective motion and self-organization of interacting particles, such as flocking and swarming, can be viewed as nonequilibrium analogues of collective dynamics in gases. Motivated by the analogy between gas mixtures and Cucker--Smale models, we introduce a polyatomic classical model and its relativistic counterpart based on the Synge energy, and analyze their large-time behavior. The relativistic formulation provides a physically consistent setting for multi-species systems where inertia and internal energy depend on temperature, as occurs in astrophysical plasmas or relativistic fluids. Using the entropy principle, we derive uniform lower bounds for temperature and establish asymptotic flocking under various communication kernels. For nearly constant interactions, flocking emerges from arbitrary initial data. The results clarify how thermodynamic effects and relativistic corrections modify the emergence of coherent motion in particle systems, bridging kinetic theory, relativistic fluid mixtures, and collective dynamics.

math.AP↗

arXiv2025

Particle, kinetic and hydrodynamic models for sea ice floes. Part I: Non-rotating floes

We introduce a comprehensive modeling framework for the dynamics of sea ice floes using particle, kinetic, and hydrodynamic approaches. Building upon the foundational work of Ha and Tadmor on the Cucker-Smale model for flocking, we derive a Vlasov-type kinetic formulation and a corresponding hydrodynamic description. The particle model incorporates essential physical properties of sea ice floes, including size, position, velocity, and interactions governed by Newtonian mechanics. By extending these principles, the kinetic model captures large-scale features through the phase-space distribution, and we also present a hydrodynamic model using the velocity moments and a suitable closure condition. In this paper, as an idea-introductory step, we assume that ice floes are non-rotating and focus on the linear velocity dynamics. Our approach highlights the role of contact forces, ocean drag effects, and conservation laws in the multiscale description of sea ice dynamics, offering a potential pathway for the improved understanding and prediction of sea ice behaviors in changing climatic conditions.

math.DS↗

arXiv2025

Quantitative relaxation dynamics from generic initial configurations in the inertial Kuramoto model

We study the relaxation dynamics of the inertial Kuramoto model toward a phase-locked state from a generic initial phase configuration. For this, we propose a sufficient framework in terms of initial data and system parameters for asymptotic phase-locking. It can be roughly stated as set of conditions such as a positive initial order parameter, a coupling strength sufficiently larger than initial frequency diameter and intrinsic frequency diameter, but less than the inverse of inertia. Under the proposed framework, generic initial configuration undergoes three dynamic stages (initial layer, condensation and relaxation stages) before it reaches a phase-locked state asymptotically. The first stage is the initial layer stage in analogy with fluid mechanics, during which the effect of the initial natural frequency distribution is dominant, compared to that of the sinusoidal coupling between oscillators. The second stage is the condensation stage, during which the order parameter increases, and at the end of which a majority cluster is contained in a sufficiently small arc. Finally, the third stage is the persistence and relaxation stage, during which the majority cluster remains stable (persistence) and the total configuration relaxes toward a phase-locked state asymptotically (relaxation). The intricate proof involves with several key tools such as the quasi-monotonicity of the order parameter (for the condensation stage), a nonlinear Grönwall inequality on the diameter of the majority cluster (for the persistence stage), and a variant of the classical Łojasiewicz gradient theorem (for the relaxation stage).

math.DS↗