arXiv subjects

Young-Pil Choi

Publications and source records attributed to Young-Pil Choi.

At least 37 records · Page 2Linked to original sources

arXiv2025

Global weak solutions to nonlinear kinetic Fokker--Planck equations in bounded domains under physical initial data

We establish the global existence of weak solutions to a nonlinear kinetic Fokker--Planck equation with degenerate diffusion, under either inflow or partial absorption-reflection boundary conditions. The novelty of our approach lies in constructing solutions under solely the physical assumptions on the initial and boundary data, namely finite mass, kinetic energy, and entropy, with no additional regularity imposed. To overcome the lack of uniform ellipticity, we develop a new compactness principle based on weighted Fisher information, which yields strong $L^1$ convergence of approximate solutions. This framework provides a robust existence theory under only the physically relevant conditions, and applies uniformly to both inflow and reflection boundary settings.

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arXiv2025

Global dynamics of damped Euler systems with exterior potentials

We study the three-dimensional isothermal Euler equations with linear damping and an exterior potential. For sufficiently large damping, we prove global well-posedness for arbitrarily large initial data by combining a parabolic comparison principle with scaled high-order energy estimates ensuring uniform density bounds. In the small-data regime with arbitrary damping, we establish global classical solutions and derive sharp algebraic decay rates via spectral analysis and frequency decomposition, and further prove their optimality under a mild non-degeneracy condition. Finally, for the pressureless damped system, we construct a weighted functional showing that solutions can blow up in finite time when the damping is insufficient, highlighting a qualitative difference from the pressured case.

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arXiv2025

Hydrodynamic limit from kinetic models with massless electrons to the ionic Euler--Poisson system

We study the derivation of ion dynamics, namely, the ionic Euler--Poisson system, from kinetic descriptions. The kinetic framework consists of the ionic Vlasov--Poisson equation coupled with either a nonlinear Fokker--Planck operator or a local alignment term. In both kinetic and fluid models, the massless electrons are assumed to be in thermodynamic equilibrium, leading to an electric potential governed by the Poisson--Boltzmann equation. The exponential nonlinearity in this semilinear elliptic problem creates significant mathematical difficulties, which we overcome by exploiting the physical structure of the system, in particular, the role of the electron velocity field hidden in the limiting equation. Our first main result establishes the hydrodynamic limit from the kinetic model to the ionic Euler--Poisson system, providing quantitative error estimates via the modulated energy method. As a second contribution, we prove the global-in-time existence of weak entropy solutions to the kinetic equations, ensuring consistency with the hydrodynamic limit framework.

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arXiv2025

Time-delayed opinion dynamics with leader-follower interactions: consensus, stability, and mean-field limits

We study a time-delayed variant of the Hegselmann-Krause opinion formation model featuring a small group of leaders and a large group of non-leaders. In this model, leaders influence all agents but only interact among themselves. At the same time, non-leaders update their opinions via interactions with their peers and the leaders, with time delays accounting for communication and decision-making lags. We prove the exponential convergence to consensus of the particle system, without imposing smallness assumptions on the delay parameters. Furthermore, we analyze the mean-field limit in two regimes: (i) with a fixed number of leaders and an infinite number of non-leaders, and (ii) with both populations tending to infinity, obtaining existence, uniqueness, and exponential decay estimates for the corresponding macroscopic models.

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arXiv2025

From kinetic mixtures to compressible two-phase flow: A BGK-type model and rigorous derivation

We propose a BGK-type kinetic model for a binary gas mixture, designed to serve as a kinetic formulation of compressible two-phase fluid dynamics. The model features species-dependent adiabatic exponents, and the relaxation operator is constructed by solving an entropy minimization problem under moments constraints. Starting from this model, we derive the compressible two-phase Euler equations via a formal Chapman--Enskog expansion and identify dissipative corrections of Navier--Stokes type. We then rigorously justify the Euler limit using the relative entropy method, establishing quantitative convergence estimates under appropriate regularity assumptions. Finally, we present numerical experiments based on an implicit-explicit Runge--Kutta method, which confirm the asymptotic preserving property and demonstrate the convergence from the BGK model to the isentropic two-phase Euler system in the hydrodynamic regime.

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arXiv2025

Large-time behavior of pressureless Euler--Poisson equations with background states

We study the large-time asymptotic behavior of solutions to the one-dimensional damped pressureless Euler-Poisson system with variable background states, subject to a neutrality condition. In the case where the background density converges asymptotically to a positive constant, we establish the convergence of global classical solutions toward the corresponding equilibrium state. The proof combines phase plane analysis with hypocoercivity-type estimates. As an application, we analyze the damped pressureless Euler--Poisson system arising in cold plasma ion dynamics, where the electron density is modeled by a Maxwell-Boltzmann relation. We show that solutions converge exponentially to the steady state under suitable a priori bounds on the density and velocity fields. Our results provide a rigorous characterization of asymptotic stability for damped Euler-Poisson systems with nontrivial background structures.

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arXiv2025

Derivation of nonlinear aggregation-diffusion equation from a kinetic BGK-type equation

This paper investigates the diffusion limit of a kinetic BGK-type equation, focusing on its relaxation to a nonlinear aggregation-diffusion equation, where the diffusion exhibits a porous-medium-type nonlinearity. Unlike previous studies by Dolbeault et al. [Arch. Ration. Mech. Anal., 186, (2007), 133-158] and Addala and Tayeb [J. Hyperbolic Differ. Equ., 16, (2019), 131-156], which required bounded initial data, our work considers initial data that need not be bounded. We develop new techniques for handling weak entropy solutions that satisfy the natural bounds associated with the kinetic entropy inequality. Our proof employs the relative entropy method and various compactness arguments to establish the convergence and properties of these solutions.

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arXiv2025

Collisionless and Decentralized Formation Control for Strings

A decentralized feedback controller for multi-agent systems, inspired by vehicle platooning, is proposed. The closed loop resulting from the decentralized control action has three distinctive features: the generation of collision-free trajectories, flocking of the system towards a consensus state in velocity, and asymptotic convergence to a prescribed pattern of distances between agents. For each feature, a rigorous dynamical analysis is provided, yielding a characterization of the set of parameters and initial configurations where collision avoidance, flocking, and pattern formation are guaranteed. Numerical tests assess the theoretical results presented.

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arXiv2024

Global smooth solutions to the irrotational Euler-Riesz system in three dimensions

This paper investigates the global dynamics of the Euler--Riesz system in three dimensions, focusing on the well-posedness and large-time behavior of solutions near equilibrium. The system generalizes classical interactions by incorporating the Riesz interactions $\nabla (-Δ)^{-σ/2}(ρ- 1)$. We show that the system admits a global smooth solution for small irrotational initial perturbations. Specifically, we establish that if the initial data is sufficiently small, the solution remains regular globally in time and decays over time at a rate dependent on $σ$.

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arXiv2024

Alignment with nonlinear velocity couplings: collision-avoidance and micro-to-macro mean-field limits

We investigate the pressureless fractional Euler-alignment system with nonlinear velocity couplings, referred to as the $p$-Euler-alignment system. This model features a nonlinear velocity alignment force, interpreted as a density-weighted fractional $p$-Laplacian when the singularity parameter $α$ exceeds the spatial dimension $d$. Our primary goal is to establish the existence of solutions for strongly singular interactions ($α\ge d$) and compactly supported initial conditions. We construct solutions as mean-field limits of empirical measures from a kinetic variant of the $p$-Euler-alignment system. Specifically, we show that a sequence of empirical measures converges to a finite Radon measure, whose local density and velocity satisfy the $p$-Euler-alignment system. Our results are the first to prove the existence of solutions to this system in multi-dimensional settings without significant initial data restrictions, covering both nonlinear ($p>2$) and linear ($p=2$) cases. Additionally, we establish global existence, uniqueness, and collision avoidance for the corresponding particle ODE system under non-collisional initial conditions, extending previous results for $1 \le p \le α+ 2$. This analysis supports our mean-field limit argument and contributes to understanding alignment models with singular communication.

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arXiv2024

Global existence of weak solutions to the nonlinear Vlasov-Fokker-Planck equation

In this paper, we study the nonlinear Vlasov-Fokker-Planck equation with fixed collision frequency. We establish the global-in-time existence of weak solutions to the equation with large initial data. Moreover, we show that our solution satisfies the conservation laws of mass, momentum, and energy, and Boltzmann's $H$-theorem.

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arXiv2024

Asymptotic synchronization of Kuramoto oscillators with time delay and non-universal interaction

We study the emergence of synchronization in the Kuramoto model on a digraph in the presence of time delays. Assuming the digraph is strongly connected, we first establish a uniform bound on the phase diameter and subsequently prove the asymptotic frequency synchronization of the oscillators under suitable assumptions on the initial configurations. In the case of an all-to-all connection, we obtain an exponential synchronization estimate. Additionally, we present numerical simulations, providing further insights into the synchronization and oscillatory behaviors of the oscillator frequencies depending on the network structure and the magnitude of the time delay.

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arXiv2024

Enhanced dissipation and temporal decay in the Euler-Poisson-Navier-Stokes equations

This paper investigates the global well-posedness and large-time behavior of solutions for a coupled fluid model in $\mathbb{R}^3$ consisting of the isothermal compressible Euler-Poisson system and incompressible Navier-Stokes equations coupled through the drag force. Notably, we exploit the dissipation effects inherent in the Poisson equation to achieve a faster decay of fluid density compared to velocities. This strategic utilization of dissipation, together with the influence of the electric field and the damping structure induced by the drag force, leads to a remarkable decay behavior: the fluid density converges to equilibrium at a rate of $(1+t)^{-11/4}$, significantly faster than the decay rates of velocity differences $(1+t)^{-7/4}$ and velocities themselves $(1+t)^{-3/4}$ in the $L^2$ norm. Furthermore, under the condition of vanishing coupled incompressible flow, we demonstrate an exponential decay to a constant state for the solution of the corresponding system, the damped Euler-Poisson system.

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arXiv2024

On well/ill-posedness for the generalized surface quasi-geostrophic equations in Hölder spaces

We establish the well/ill-posedness theories for the inviscid $α$-surface quasi-geostrophic ($α$-SQG) equations in Hölder spaces, where $α= 0$ and $α= 1$ correspond to the two-dimensional Euler equation in the vorticity formulation and SQG equation of geophysical significance, respectively. We first prove the local-in-time well-posedness of $α$-SQG equations in $C([0,T);C^{0,β}(\mathbb{R}^2))$ with $β\in (α,1)$ for some $T>0$. We then analyze the strong ill-posedness in $C^{0,α}(\mathbb{R}^2)$ constructing smooth solutions to the $α$-SQG equations that exhibit $C^{0,α}$--norm growth in a short time. In particular, we develop the nonexistence theory for $α$-SQG equations in $C^{0,α}(\mathbb{R}^2)$.

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arXiv2024

Weak-strong uniqueness and high-friction limit for Euler-Riesz systems

In this work we employ the relative energy method to obtain a weak-strong uniqueness principle for a Euler-Riesz system, as well as to establish its convergence in the high-friction limit towards a gradient flow equation. The main technical challenge in our analysis is addressed using a specific case of a Hardy-Littlewood-Sobolev inequality for Riesz potentials.

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arXiv2024

Global existence and asymptotic stability for the Toner-Tu model of flocking

This paper deals with the Toner-Tu (TT) model, which is a hydrodynamic model describing the collective motion of numerous self-propelled agents. We analytically study the global-in-time well-posedness of the TT model near the steady-state solution in the ordered phase. We also show the large-time behavior of solutions showing that the steady-state solution is polynomially stable in a Sobolev space in the sense that solutions that are initially close to that steady state converge to that at least polynomially fast as time tends to infinity. Moreover, we investigate the variant of the TT model which describes the dynamics of the actin filament.

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arXiv2024

Critical thresholds in pressureless Euler--Poisson equations with background states

We investigate the critical threshold phenomena in a large class of one dimensional pressureless Euler--Poisson (EP) equations, with non-vanishing background states. First, we establish local-in-time well-posedness in proper regularity spaces, which are adapted for a certain \textit{neutrality condition} to hold. The neutrality condition is shown to be necessary: we construct smooth solutions that exhibit instantaneous failure of the neutrality condition, which in turn yields non-existence of solutions, even locally in time, in the classical Sobolev spaces $H^s({\mathbb R})$, $s \geq 2$. Next, we study the critical threshold phenomena in the neutrality-condition-satisfying pressureless EP systems, where we distinguish between two cases. We prove that in the case of attractive forcing, the neutrality condition can further restrict the sub-critical region into its borderline, namely -- the sub-critical region is reduced to a single line in the phase plane. We then turn to provide a rather definitive answer for the critical thresholds in the case of repulsive EP systems with variable backgrounds. As an application, we analyze the critical thresholds for the damped EP system for cold plasma ion dynamics, where the density of electrons is given by the \textit{Maxwell--Boltzmann relation}.

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arXiv2024

Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming

We consider a one-dimensional hydrodynamic model featuring nonlocal attraction-repulsion interactions and singular velocity alignment. We introduce a two-velocity reformulation and the corresponding energy-type inequality, in the spirit of the Bresch-Desjardins estimate. We identify a dependence between the communication weight and interaction kernel and between the pressure and viscosity term allowing for this inequality to be uniform in time. It is then used to study long-time asymptotics of solutions.

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