arXiv subjects

Dowan Koo

Publications and source records attributed to Dowan Koo.

12 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

Conditional hypocoercivity for nonlinear kinetic Fokker--Planck equations

We investigate the long-time behaviour of nonlinear kinetic Fokker--Planck equations with porous medium diffusion in a non-perturbative setting. Under a priori conditional bounds on macroscopic quantities, we establish exponential convergence to equilibrium in $L^1$. These bounds are automatically satisfied if the initial data is trapped between two global equilibrium profiles. Our approach combines the entropy-entropy dissipation structure and some techniques from $L^2$-hypocoercivity.

math.AP↗

arXiv2026

Hydrodynamic limit from nonlinear Fokker--Planck to barotropic Euler equations

The hydrodynamic limit to the barotropic Euler equations, including power-law pressure $P(\rho)=\rho^\gamma$, for a kinetic nonlinear Fokker--Planck equation with degenerate diffusion is established. This extends the well-known result of the derivation of isothermal Euler equations via Fokker--Planck equation with linear diffusion. We establish the asymptotic analysis using the relative entropy method by quantifying error estimates for pressures and employing the generalized Log-Sobolev inequality for degenerate diffusion.

math.AP↗

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 $(\lambda,\Lambda)$-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 $(\lambda,\Lambda)$-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

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.

math.AP↗

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.

math.AP↗

arXiv2025

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

In this paper, we establish the global existence of Lagrangian solutions to the ionic Vlasov--Poisson system under mild integrability assumptions on the initial data. Our approach involves proving the well-posedness of the Poisson--Boltzmann equation for densities in $L^p$ with $p>1$, introducing a novel decomposition technique that ensures uniqueness, stability, and improved bounds for the thermalized electron density. Using this result, we construct global-in-time Lagrangian solutions while demonstrating that the energy functional remains uniformly bounded by its initial value. Additionally, we show that renormalized solutions coincide with Lagrangian solutions, highlighting the transport structure of the system, and prove that renormalized solutions coincide with weak solutions under additional integrability assumptions.

math.AP↗

arXiv2024

Global Mild Solutions to a BGK Model for Barotropic Gas Dynamics

We establish global existence of mild solutions to the BGK model proposed by Bouchut [J. Stat. Phys., 95, (1999), 113--170] under the minimal assumption of finite kinetic entropy initial data. Moreover we rigorously derive a kinetic entropy inequality, which combined with the theory developed by Berthelin and Vasseur [SIAM J. Math. Anal., 36, (2005), 1807--1835] leads to the hydrodynamic limit to the barotropic Euler equations. The main tools employed in the analysis are stability estimates for the Maxwellian and a velocity averaging lemma.

math.AP↗

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}.

math.AP↗

arXiv2022

A gradient flow for the Porous Medium Equations with Dirichlet boundary conditions

We consider the gradient flow structure of the porous medium equations with non-negative constant Dirichlet boundary conditions. We construct weak solutions to the equations via the minimizing movement scheme by considering an entropy functional with respect to $Wb_2$ distance, which is a modified Wasserstein distance introduced by Figalli and Gigli [J. Math. Pures Appl. 94, (2010), pp. 107-130]. Furthermore, the constructed solutions are characterized as curves of maximal slope in a suitable sense.

math.AP↗

arXiv2022

A novel approach for wafer defect pattern classification based on topological data analysis

In semiconductor manufacturing, wafer map defect pattern provides critical information for facility maintenance and yield management, so the classification of defect patterns is one of the most important tasks in the manufacturing process. In this paper, we propose a novel way to represent the shape of the defect pattern as a finite-dimensional vector, which will be used as an input for a neural network algorithm for classification. The main idea is to extract the topological features of each pattern by using the theory of persistent homology from topological data analysis (TDA). Through some experiments with a simulated dataset, we show that the proposed method is faster and much more efficient in training with higher accuracy, compared with the method using convolutional neural networks (CNN) which is the most common approach for wafer map defect pattern classification. Moreover, our method outperforms the CNN-based method when the number of training data is not enough and is imbalanced.

cs.LG↗

arXiv2021

One dimensional consensus based algorithm for non-convex optimization

We analyze the consensus based optimization method proposed by Pinnau et al.(2017) in one dimension. We rigorously provide a quantitative error estimate between the consensus point and global minimizer of a given objective function. Our analysis covers general objective functions; we do not require any structural assumption on the objective function.

math.OC↗