arXiv subjects

Young-Pil Choi

Publications and source records attributed to Young-Pil Choi.

At least 19 recordsLinked to original sources

arXiv2026

Global existence of Lagrangian solutions to the ionic Vlasov--Poisson system

In this paper, we study the Cauchy problem of the ionic Vlasov--Poisson (IVP) system for a broad class of initial data. To this end, we establish well-posedness and stability estimates for the Poisson--Boltzmann equation $-ΔΦ= ρ-e^Φ$ provided $ρ\in L_+^p$ with $p>1$, whereas previous approaches required $p>\frac{d}{2}$. Building upon these results, we construct global-in-time renormalized/Lagrangian solutions to the IVP system.

math.AP↗

arXiv2026

Regularity theory and low Mach number limit for the fractional Euler-alignment system

We study the compressible Euler-alignment system with pressure under a singular pressure scaling, where the hypersingular communication weight induces a fractional alignment operator of order $2α$, $0<α<1$. The scaling corresponds to a large-time and small-velocity regime and leads to a low Mach number problem in which the density is forced to remain close to a constant state. Our main result is a uniform regularity theory for this scaled pressure system. We establish uniform estimates with respect to the scaling parameter and construct global strong solutions near the constant state. A key feature of the analysis is that, in the low-order fractional regime $0<α\le\frac12$, the estimates close under the lower Sobolev condition $s>\frac d2+1-2α$, gaining $α$ derivatives over the threshold $s>\frac d2 + 1 - α$ arising from a direct use of the fractional alignment dissipation. This is achieved by combining refined commutator estimates for the singular alignment operator with the density dissipation induced by the pressure scaling. As an application of the uniform estimates, we justify the low Mach number limit toward the incompressible Navier--Stokes system with fractional dissipation. For general small, possibly ill-prepared initial data, a Helmholtz decomposition combined with dispersive estimates for the acoustic component yields subsequential strong convergence locally in space-time to a distributional solution of the limiting system. For well-prepared initial data, a relative-energy argument further identifies the limit with a prescribed sufficiently regular solution and yields global strong convergence in the fractional dissipation norm.

math.AP↗

arXiv2026

Global existence and time decay for a bipolar Euler-Poisson system with one pressureless and undamped fluid

We study the Cauchy problem for a three-dimensional bipolar Euler--Poisson system in which one fluid is pressureless and undamped, while the other is subject to momentum relaxation. For sufficiently small smooth perturbations of a constant equilibrium, we prove the global existence and uniqueness of smooth solutions under an irrotationality assumption on the initial velocity of the pressureless fluid, together with algebraic time-decay estimates. The main difficulty is that the velocity of the pressureless fluid is dissipated only indirectly through the Poisson coupling, and this mechanism degenerates strongly at high frequencies, leading to a regularity-loss structure. We overcome this difficulty by combining refined Green-function estimates, a low--middle--high frequency decomposition, and high-order nonlinear energy estimates adapted to the asymmetric regularity hierarchy. The result establishes a global small-data theory for this asymmetric regime, in which pressure and damping are simultaneously absent from the same fluid.

math.AP↗

arXiv2026

Global mild solutions and the semiclassical limit for the Fermi--Dirac BGK model

We prove global existence of mild solutions to the spatially inhomogeneous Fermi--Dirac BGK equation for arbitrary-size Pauli-admissible initial data with finite mass and kinetic energy, allowing both vacuum and locally saturated zero-temperature states. The construction relies on a moment-compatible regularization of the local equilibrium that preserves the Pauli bound and uniform moment control while remaining consistent up to the saturation boundary. The solutions satisfy mass, momentum, and energy conservation and, under an additional finite spatial second-moment assumption, an entropy-gap H-theorem on the full Pauli interval. Under uniform moment and entropy bounds, we further prove that, after extraction of a subsequence, quantum mild solutions converge strongly in phase space, uniformly on bounded time intervals, to a mild solution of the classical BGK equation.

math.AP↗

arXiv2026

Finite-energy weak solutions and relaxation for a compressible kinetic--fluid system with locally averaged Brinkman force

We study a kinetic--fluid system in which a Vlasov or Vlasov--Fokker--Planck equation is coupled to the compressible Navier--Stokes equations with density-dependent viscosities through a locally averaged Brinkman force. The averaging is chosen in a conservative form so that the coupled system preserves the total momentum and satisfies a natural energy-dissipation balance, while avoiding the pointwise evaluation of a possibly rough fluid velocity. The first main result is a conditional exponential relaxation estimate for sufficiently regular solutions. The estimate is proved under positive and negative density moment bounds and a Muckenhoupt $\calA_2$ condition on a power of the density, which replace uniform pointwise upper and lower bounds. The proof combines a modulated energy and hypocoercivity analysis with a compensating functional for the density fluctuation. A key ingredient is a weighted Calderón--Zygmund estimate for the associated elliptic corrector, which allows us to control the viscous contribution under the $\calA_2$ condition. The second main result concerns global finite-energy Bresch--Desjardins entropy weak solutions in admissible low-dimensional regimes. Using the additional entropy structure associated with the density-dependent viscosities, we verify the density assumptions required by the conditional relaxation theorem and obtain exponential relaxation for the weak solutions. In the diffusionless case, the particle distribution aligns toward a mono-kinetic state, while in the diffusive case the relaxation is towards a Maxwellian equilibrium.

math.AP↗

arXiv2026

Incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation

We derive the incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation with Bose-Einstein or Fermi-Dirac statistics. The model has a self-consistent collision structure, with the local density acting as the collision frequency and the bulk velocity and temperature determined by nonlinear quantum-weighted moments of the distribution. We work near a global quantum equilibrium under the diffusive scaling and keep the quantum parameter fixed. Uniform estimates with respect to the Knudsen number yield strong microscopic relaxation and identify the limiting infinitesimal quantum equilibrium. Using the local conservation laws, we prove the incompressibility condition, the Boussinesq relation, and strong compactness of the divergence-free velocity component and a quantum-adapted thermal mode, while the acoustic modes vanish locally by a dispersive estimate. The limiting viscous stress tensor and heat flux are identified by solving auxiliary equations for the linearized quantum Fokker-Planck operator and by expanding the local quantum equilibrium manifold. The resulting incompressible Navier-Stokes-Fourier system retains the effect of quantum statistics through its normalization constants and transport coefficients.

math.AP↗

arXiv2026

Nonlinear quantum Fokker-Planck equation near equilibrium

We investigate a nonlinear quantum Fokker--Planck equation with self-consistent collision frequency, bulk velocity, and temperature. In contrast to quantum Fokker--Planck equations with prescribed diffusion and friction coefficients, the macroscopic quantities are nonlinear functionals of the distribution function. The equation preserves mass, momentum, and kinetic energy, admits a quantum entropy dissipation structure, and propagates the Pauli admissible range in the fermionic case. Its collision operator is also formally connected to the quantum Landau equation. For the Cauchy problem in the three-dimensional whole space, we prove the global-in-time existence and uniqueness of strong solutions near a global quantum equilibrium. The proof is based on a perturbative macro--micro energy method that combines microscopic coercivity, estimates for nonlinear velocity moments, and a macroscopic dissipation argument. We further establish the propagation of nonnegativity and the fermionic Pauli upper bound. Under an additional negative Sobolev assumption on the initial perturbation, we obtain algebraic decay rates toward equilibrium.

math.AP↗

arXiv2026

Finite-time breakdown of the Euler-alignment system for supercritical initial data

We study finite-time breakdown of classical solutions to the Euler-alignment system through the degeneration of the associated Lagrangian flow. This approach allows us to characterize singularity formation in terms of the loss of local invertibility of the flow and the resulting concentration of density along characteristics. For the case of constant communication kernels, we derive an explicit formula for the flow and obtain an exact pointwise breakdown criterion in arbitrary dimension. In two dimensions, this criterion admits a closed-form reformulation in terms of the symmetric part of the initial velocity gradient and the initial vorticity. For general non-constant kernels, we derive sufficient conditions for finite-time degeneracy by combining a leading compressive mechanism with perturbative control of the nonlocal remainder. These conditions provide quantitative supercritical breakdown criteria in arbitrary dimension, complementing the existing subcritical global-regularity theory for multidimensional Euler-alignment systems.

math.AP↗

arXiv2026

Coupled Vlasov and non-Newtonian fluid dynamics: existence and large-time behavior

We study a coupled kinetic-non-Newtonian fluid system on the periodic domain ${\mathbb T}^3$, where particles evolve by a Vlasov equation and interact with an incompressible power-law fluid through a drag force. We prove the global existence of weak solutions for all $p > \frac{8}{5}$, where $p > 1$ denotes the power-law exponent of the fluid's stress-strain relation. Under an additional uniform boundedness assumption on the particle density, we also establish large-time decay of a modulated energy functional measuring deviation from velocity alignment. The decay rate is algebraic when $p > 2$ and exponential when $\frac{6}{5} \le p \le 2$, reflecting the role of fluid dissipation in the large-time dynamics.

math.AP↗

arXiv2026

Quantitative light-particle limit for the Vlasov-Fokker-Planck-Navier-Stokes system

We investigate the hydrodynamic limit of the Vlasov-Fokker-Planck-Navier-Stokes system in the light particle regime, where the particle relaxation takes place on a singularly fast time scale. Using a relative entropy method adapted to this scaling, we develop the first quantitative convergence theory for the light particle limit. Our analysis yields explicit rates for the convergence of both the kinetic distribution and the fluid velocity, extending the qualitative compactness-based result of Goudon, Jabin, and Vasseur [Indiana Univ. Math. J., 53, (2004), 1495-1515]. Moreover, we derive refined convergence estimates for the macroscopic density and fluid velocity in negative Sobolev spaces, consistent with the formally optimal rates predicted by the Hilbert expansion. The results apply to both the torus and the whole space, providing a unified quantitative description of the light particle hydrodynamic limit.

math.AP↗

arXiv2026

Global large smooth solutions and overdamped limits for the damped isothermal Euler-Poisson system

We consider the isothermal Euler-Poisson system with linear damping on a periodic domain in the large damping regime. For arbitrarily large smooth initial data with density bounded away from vacuum, we prove the global-in-time existence of smooth solutions. The argument is based on a large-damping bootstrap scheme, a modified density estimate revealing hidden parabolic dissipation, top-order weighted cancellations, and a comparison with an auxiliary drift-diffusion--Poisson system. We further prove exponential relaxation to the homogeneous equilibrium and establish a large-data overdamped limit as the damping coefficient tends to infinity. In the slow time scale, the density converges quantitatively to the large smooth solution of the drift-diffusion--Poisson system with the same initial density. After subtracting a fast initial layer from the rescaled flux, the flux converges quantitatively to the corresponding drift-diffusion flux.

math.AP↗

arXiv2026

A unified relative entropy framework for macroscopic limits of Vlasov--Fokker--Planck equations

We develop a unified relative entropy framework for macroscopic limits of kinetic equations with Riesz-type interactions and Fokker-Planck relaxation. Our analysis covers three prototypical singular regimes: the diffusive limit leading to a drift-diffusion equation, the high-field limit yielding the aggregation equation in the repulsive regime, and the strong magnetic field limit producing a generalized surface quasi-geostrophic equation. The method combines entropy dissipation, Fisher-information control, and modulated interaction energies into a robust stability theory yielding both strong and weak convergence results. For the strong convergence, we establish quantitative relative entropy estimates toward macroscopic limits under well-prepared initial data, extending the scope of the method to settings where nonlocal forces and singular scalings play a decisive role. For the weak convergence, our approach captures three complementary phenomena: in the diffusive regime, it yields sharper quantitative estimates in weak topologies consistent with the formally optimal scaling; in the high-field regime, it propagates bounded Lipschitz stability for a class of mildly prepared initial data, even when the relative entropy diverges with respect to the singular scaling parameter; and in the strong magnetic field regime, it provides quantitative weak estimates, including bounded Lipschitz control of the rescaled momentum and negative Sobolev control of the density. This broader perspective shows that relative entropy provides not only a tool for strong convergence, but also a mechanism for treating low-regularity and mildly prepared regimes. The analysis highlights the unifying role of relative entropy in connecting microscopic dissipation with both strong and weak macroscopic convergence.

math.AP↗

arXiv2026

Lagrangian formulation and Eulerian closure in alignment dynamics

We investigate a continuum Lagrangian $p$-alignment system given by a nonlocal mean-field system of ordinary differential equations for interacting agents with weak initial data. We first establish global well-posedness of the Lagrangian dynamics and derive quantitative flocking estimates. We next construct Eulerian variables from the possibly non-injective Lagrangian flow via pushforward and disintegration, which leads to an Euler--Reynolds--alignment system featuring a nonnegative Reynolds stress and, for $p>2$, a nonlinear defect force induced by microscopic velocity fluctuations. Assuming only heavy-tailed interaction, we then show that these defect terms vanish asymptotically, leading to asymptotic mono-kinetic closure in the long-time limit. In the linear case $p=2$, we further obtain global weak solutions to the Euler--alignment system, including a sharp one-dimensional critical-threshold characterization and a global result in higher dimensions under a large-coupling condition. Finally, we establish a uniform-in-time mean-field stability estimate for the particle Cucker--Smale system in the linear regime and deduce uniform-in-time convergence toward the mono-kinetic Eulerian limit; for general $p\ge2$, we also obtain a finite-time mean-field convergence result toward the associated kinetic/Lagrangian alignment dynamics.

math.AP↗

arXiv2026

Uniform-in-time propagation of chaos and bifurcation in two-type adhesion systems

We study a nonlocal adhesion model for two interacting tumor cell phenotypes, combining diffusion, pairwise interactions, and random phenotypic switching. The system admits a microscopic diffusion--jump particle description whose mean-field limit is a nonlinear McKean--Vlasov equation on a product space encoding position and internal state. We first establish uniform-in-time propagation of chaos in the weak-interaction regime using a coupling approach that combines reflection coupling for the diffusion with an optimal coupling of the spin-flip dynamics. As a byproduct, we obtain exponential long-time contraction for the nonlinear McKean--Vlasov equation in the first-order Wasserstein distance, implying uniqueness of the stationary distribution. We also investigate the complementary regime of strong interactions, where the homogeneous equilibrium may lose stability through a bifurcation mechanism.

math.AP↗

arXiv2025

A modified Consensus-Based Optimization model: consensus formation and uniform-in-time propagation of chaos

We introduce a modified Consensus-Based Optimization model that admits a fully unified and rigorous analysis of its finite-particle dynamics, the associated McKean--Vlasov equation, and their optimization behavior under a single set of structural framework. The key ingredient is a regularized Gibbs weight that stabilizes the consensus point and avoids degeneracies present in the classical formulation, eliminating the need for cutoffs, rescaling, or boundedness assumptions on the objective function. Our first main result establishes large-time consensus for the particle system: when the drift exceeds an explicit threshold, all particles converge exponentially to a common random limit that concentrates near the global minimizer. Our second result proves uniform-in-time propagation of chaos, providing quantitative and dimension-free convergence of the empirical measure to the McKean--Vlasov dynamics. Finally, we show that the mean-field system reaches deterministic consensus and that its consensus point approaches the global minimizer in the regime of highly concentrated Gibbs weights. Together, these results yield a unified and internally consistent theoretical framework for consensus-based optimization under substantially relaxed regularity assumptions on the objective function.

math.PR↗

arXiv2025

Exponential and algebraic decay in Euler--alignment system with nonlocal interaction forces

We investigate the large-time behavior of the pressureless Euler system with nonlocal velocity alignment and interaction forces, with the aim of characterizing the asymptotic convergence of classical solutions under general interaction potentials $W$ and communication weights. We establish quantitative convergence in three settings. In one dimension with $(λ,Λ)$-convex potentials, i.e., potentials satisfying uniform lower and upper quadratic bounds, bounded communication weights yield exponential decay, while weakly singular ones lead to sharp algebraic rates. For the Coulomb--quadratic potential $W(x)=-|x|+\frac12 |x|^2$, we prove exponential convergence for bounded communication weights and algebraic upper bounds for singular communication weights. In a multi-dimensional setting with uniformly $(λ,Λ)$-convex potentials, we show exponential decay for bounded weights and improved algebraic decay for singular ones. In all cases, the density converges (up to translation) to the minimizer of the interaction energy, while the velocity aligns to a uniform constant. A unifying feature is that the convergence rate depends only on the local behavior of communication weights: bounded kernels yield exponential convergence, while weakly singular ones produce algebraic rates. Our results thus provide a comprehensive description of the asymptotic behavior of Euler--alignment dynamics with general interaction potentials.

math.AP↗

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.

math.AP↗

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.

math.AP↗