arXiv subjects

Jin Woo Jang

Publications and source records attributed to Jin Woo Jang.

At least 19 recordsLinked to original sources

arXiv2026

Long-Time Dynamics of the 3D Vlasov-Maxwell System with Boundaries

We construct global-in-time classical solutions to the nonlinear Vlasov--Maxwell system in a three-dimensional half-space beyond the vacuum scattering regime in a physical setting of the solar wind model. Our approach combines the construction of stationary solutions to the associated boundary-value problem with a proof of their asymptotic dynamical stability in $L^\infty$ under small perturbations, providing a new framework for understanding long-time wave-particle interactions in the presence of boundaries and interacting magnetic fields. A key difficulty is that the transport dynamics remains coupled on arbitrarily long time scales to a wave component whose sharp $t^{-1}$ pointwise decay is neither integrable in time nor improved by the boundary. We resolve this obstruction by establishing a new decay mechanism that converts the spatial localization of particle--field interactions into temporal decay through the light-cone geometry of wave propagation. This enables us to close the nonlinear field--particle feedback and establish sharp $t^{-1}$ decay for both the particle distribution and the electromagnetic fields. To the best of our knowledge, this work presents the first construction of asymptotically stable non-vacuum steady states under general perturbations in the full three-dimensional nonlinear Vlasov--Maxwell system.

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arXiv2026

Local Existence and Finite-Time Singularity Formation in the Vlasov-Poisson-Isotropic Landau System

The isotropic Landau (Coulomb) operator was introduced in kinetic theory by Krieger and Strain (Comm. Partial Differential Equations, 2012). In this work, we study the spatially inhomogeneous Vlasov--Poisson--isotropic Landau system. We first establish a local--in--time existence theory for the Cauchy problem: for initial data satisfying a suitable smallness condition in an appropriate norm, there exists a non--negative solution on a time interval $[0,T]$, where the lifespan $T$ depends on the size of the initial data. Beyond the local theory, we investigate a mechanism that may lead to the breakdown of global existence. We show that finite--time singularity formation can occur in the gravitationally attractive case, provided that the weak solution satisfies certain a priori regularity and decay assumptions, the initial gravitational field energy exceeds the kinetic energy, and the resulting energy gap dominates the diffusive effect of the collision operator. As a consequence, if the solution is further assumed to belong to a suitable measure space up to the maximal existence time, it collapses to a single point in physical space at that time. The proof of finite--time singularity formation is based on deriving an upper bound for the second spatial moment, which becomes negative in finite time.

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arXiv2026

Steady Solutions to the Relativistic Boltzmann Equation in a Slab

We study steady solutions to the relativistic Boltzmann equation with hard-sphere interactions in a slab geometry. Under a spatial symmetry assumption in the transverse variables $x_2$ and $x_3$, the problem reduces to a one-dimensional spatial slab $x_1 \in [0,1]$ while retaining full three-dimensional momentum dependence. For non-negative inflow boundary conditions prescribed at $x_1=0$ and $x_1=1$, we prove the existence and uniqueness of a stationary solution in a weighted $L^1_p L^\infty_{x_1}$ framework, together with exponential decay in momentum. Our analysis treats the full slab domain and does not rely on any smallness assumption on the slab width. We establish sharp coercivity and continuity estimates for the collision frequency, together with weighted convolution and pointwise bounds for the nonlinear gain term. These estimates generate and propagate a $(-Δ_p)^{-1}$-type regularity within the solution framework, which plays a crucial role in the existence and uniqueness argument. In addition, we obtain uniform weighted integrability of the solution over arbitrary two-dimensional hyperplanes through the origin. This hyperplane estimate is derived as a genuinely a posteriori regularity property, without imposing any a priori hyperplane bounds, and follows from a Lorentz-invariant geometric reduction.

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arXiv2025

Optimal Convergence Estimate of the Limit from Inverse Power Potential to Hard Sphere Boltzmann Equation

The inverse power potential $U(r)=r^{-1/s}, 0<s<1$, generates the Boltzmann kernel $B^{s}=|v-v_*|^{1-4s} b_s(θ)$ with an angular singularity as $θ\to 0$. Jang-Kepka-Nota-Velázquez (2023) proved the limit $B^{s}\to \frac14|v-v_*|$ as $s\to 0$, as well as weak convergence of solutions based on this kernel convergence. In this work we establish the following sharp quantitative estimate: $$ |b_s(θ)-\tfrac14| \le C\, s\,θ^{-2-2s}. $$ In particular, this sharp estimate yields the optimal $O(s)$ convergence rate for solutions of the homogeneous Boltzmann equation with large initial data in suitable Sobolev spaces; i.e., for any $t\in[0,T]$, we have $$f^s(t)=f^0(t)+O(s),$$ quantified by the $L^1_k$ norm for $k\ge 2.$

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arXiv2025

Quantitative pointwise estimates of the cooling process for inelastic Boltzmann equation

In this paper, we study the homogeneous inelastic Boltzmann equation for hard spheres. We first prove that the solution $f(t,v)$ is bounded pointwise from above by $C_{f_0}\langle t \rangle^3$ and establish that the cooling time is infinite $T_c = +\infty$ under the condition $f_0 \in L^1_2 \cap L^{\infty}_{s}$ for $s > 2$. Away from zero velocity, we further prove that $f(t,v)\leq C_{f_0, |v|} \langle t \rangle$ for $v \neq 0$ at any time $t > 0$. This time-dependent pointwise upper bound is natural in the cooling process, as we expect the density near $v = 0$ to grow rapidly. We also establish an upper bound that depends on the coefficient of normal restitution constant, $α\in (0,1]$. This upper bound becomes constant when $α= 1$, restoring the known upper bound for elastic collisions [8]. Consequently, through these results, we obtain Maxwellian upper bounds on the solutions at each time.

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arXiv2025

Deep learning-based moment closure for multi-phase computation of semiclassical limit of the Schrödinger equation

We present a deep learning approach for computing multi-phase solutions to the semiclassical limit of the Schrödinger equation. Traditional methods require deriving a multi-phase ansatz to close the moment system of the Liouville equation, a process that is often computationally intensive and impractical. Our method offers an efficient alternative by introducing a novel two-stage neural network framework to close the $2N\times 2N$ moment system, where $N$ represents the number of phases in the solution ansatz. In the first stage, we train neural networks to learn the mapping between higher-order moments and lower-order moments (along with their derivatives). The second stage incorporates physics-informed neural networks (PINNs), where we substitute the learned higher-order moments to systematically close the system. We provide theoretical guarantees for the convergence of both the loss functions and the neural network approximations. Numerical experiments demonstrate the effectiveness of our method for one- and two-dimensional problems with various phase numbers $N$ in the multi-phase solutions. The results confirm the accuracy and computational efficiency of the proposed approach compared to conventional techniques.

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arXiv2025

On the axially symmetric solutions to the spatially homogeneous Landau equation

In this paper, we consider the spatially homogeneous Landau equation, which is a variation of the Boltzmann equation in the grazing collision limit. For the Landau equation for hard potentials in the style of Desvillettes-Villani (Comm. Partial Differential Equations, 2000), we provide the proof of the existence of axisymmetric measure-valued solution for any axisymmetric $\mathcal{P}_p(\mathbb{R}^3)$ initial profile for any $p\ge 2$. Moreover, we prove that if the initial data is not a single Dirac mass, then the solution instantaneously becomes analytic for any time $t>0$ in the hard potential case. In the soft potential and the Maxwellian molecule cases, we show that there are no solutions whose support is contained in a fixed line even for any given line-concentrated data.

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arXiv2024

Compactness and existence theory for a general class of stationary radiative transfer equations

In this paper, we study the steady-states of a large class of stationary radiative transfer equations in a $C^1$ convex bounded domain. Namely, we consider the case in which both absorption-emission and scattering coefficients depend on the local temperature $T$ and the radiation frequency $ν.$ The radiative transfer equation determines the temperature of the material at each point. The main difficulty in proving existence of solutions is to obtain compactness of the sequence of integrals along lines that appear in several exponential terms. We prove a new compactness result suitable to deal with such a non-local operator containing integrals on a line segment. On the other hand, to obtain the existence theory of the full equation with both absorption and scattering terms we combine the compactness result with the construction of suitable Green functions for a class of non-local equations.

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arXiv2023

On the temperature distribution of a body heated by radiation

In this paper, we study the temperature distribution of a body when the heat is transmitted only by radiation. The heat transmitted by convection and conduction is ignored. We consider the stationary radiative transfer equation in the local thermodynamic equilibrium. We prove that the stationary radiative transfer equation coupled with the non-local temperature equation is well-posed in a generic case when emission-absorption or scattering of interacting radiation is considered. The emission-absorption and the scattering coefficients are assumed to be general and they can depend on the frequency of radiation. We also establish an entropy production formula of the system, which is used to prove the uniqueness of solutions for an incoming radiation with constant temperature.

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arXiv2023

Moments creation for the inelastic Boltzmann equation for hard potentials without angular cutoff

This paper is concerned with the inelastic Boltzmann equation without angular cutoff. We work in the spatially homogeneous case. We establish the global-in-time existence of measure-valued solutions under the generic hard potential long-range interaction on the collision kernel. In addition, we provide a rigorous proof for the creation of polynomial moments of the measure-valued solutions, which is a special property that can only be expected from hard potential collisional cross-sections. The proofs rely crucially on the establishment of a refined Povzner-type inequality for the inelastic Boltzmann equation without angular cutoff. The class of initial data that we require is general in the sense that we only require the boundedness of $(2+κ)$-moment for $κ>0$ and do not assume any entropy bound.

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arXiv2022

Vanishing angular singularity limit to the hard-sphere Boltzmann equation

In this note we study Boltzmann's collision kernel for inverse power law interactions $U_s(r)=1/r^{s-1}$ for $s>2$ in dimension $ d=3 $. We prove the limit of the non-cutoff kernel to the hard-sphere kernel and give precise asymptotic formulas of the singular layer near $θ\simeq 0$ in the limit $ s\to \infty $. Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions.

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arXiv2022

Kinetic Models for Semiflexible Polymers in a Half-plane

Based on a general discrete model for a semiflexible polymer chain, we introduce a formal derivation of a kinetic equation for semiflexible polymers in the half-plane via a continuum limit. It turns out that the resulting equation is the kinetic Fokker-Planck-type equation with the Laplace-Beltrami operator under a non-local trapping boundary condition. We then study the well-posedness and the long-chain asymptotics of the solutions of the resulting equation. In particular, we prove that there exists a unique measure-valued solution for the corresponding boundary value problem. In addition, we prove that the equation is hypoelliptic and the solutions are locally Hölder continuous near the singular boundary. Finally, we provide the asymptotic behaviors of the solutions for large polymer chains.

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arXiv2022

Frequency multiplier estimates for the linearized relativistic Boltzmann operator without angular cutoff

This paper is concerned with the relativistic Boltzmann equation without angular cutoff. The non-cutoff theory for the relativistic Boltzmann equation has been rarely studied even under a smallness assumption on the initial data due to the lack of understanding of the spectrum and the need for coercivity estimates on the linearized collision operator. Namely, it is crucial to obtain the sharp asymptotics for the frequency multiplier to obtain this coercivity that has never been established before. In this paper, we prove the sharp asymptotics for the frequency multiplier for a general relativistic scattering kernel without angular cutoff. As a consequence of our calculations, we further explain how the well-known change of variables $p' \to p$ is not well defined in the special relativistic context.

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arXiv2022

Asymptotic Stability of the Relativistic Boltzmann Equation without Angular Cut-off

This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness, and asymptotic stability for solutions nearby the relativistic Maxwellian. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Je$\dot{\text{z}}$ewska (Comm. Math. Phys. \textbf{115}(4):607--629, 1985), and our assumptions include the case of Israel particles (J. Math. Phys. \textbf{4}:1163--1181, 1963). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. The coercivity estimates that are needed rely crucially on the sharp asymptotics for the frequency multiplier that has not been previously established. We further derive the relativistic analogue of the Carleman dual representation for the Boltzmann collision operator. This resolves the open question of perturbative global existence and uniqueness without the Grad's angular cut-off assumption.

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arXiv2021

Sobolev Training for Physics Informed Neural Networks

Physics Informed Neural Networks (PINNs) is a promising application of deep learning. The smooth architecture of a fully connected neural network is appropriate for finding the solutions of PDEs; the corresponding loss function can also be intuitively designed and guarantees the convergence for various kinds of PDEs. However, the rate of convergence has been considered as a weakness of this approach. This paper proposes Sobolev-PINNs, a novel loss function for the training of PINNs, making the training substantially efficient. Inspired by the recent studies that incorporate derivative information for the training of neural networks, we develop a loss function that guides a neural network to reduce the error in the corresponding Sobolev space. Surprisingly, a simple modification of the loss function can make the training process similar to \textit{Sobolev Training} although PINNs is not a fully supervised learning task. We provide several theoretical justifications that the proposed loss functions upper bound the error in the corresponding Sobolev spaces for the viscous Burgers equation and the kinetic Fokker--Planck equation. We also present several simulation results, which show that compared with the traditional $L^2$ loss function, the proposed loss function guides the neural network to a significantly faster convergence. Moreover, we provide the empirical evidence that shows that the proposed loss function, together with the iterative sampling techniques, performs better in solving high dimensional PDEs.

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arXiv2021

LTE and Non-LTE Solutions in Gases Interacting with Radiation

In this paper, we study a class of kinetic equations describing radiative transfer in gases which include also the interaction of the molecules of the gas with themselves. We discuss several scaling limits and introduce some Euler-like systems coupled with radiation as an aftermath of specific scaling limits. We consider scaling limits in which local thermodynamic equilibrium (LTE) holds, as well as situations in which this assumption fails (non-LTE). The structure of the equations describing the gas-radiation system is very different in the LTE and non-LTE cases. We prove the existence of stationary solutions with zero velocities to the resulting limit models in the LTE case. We also prove the non-existence of a stationary state with zero velocities in a non-LTE case.

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arXiv2021

Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus

Although the nuclear fusion process has received a great deal of attention in recent years, the amount of mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The authors hope that this work is a step towards a more generalized work on the three-dimensional Tokamak structure. The highlight of this work is the physical assumptions on the external magnetic potential well which remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system.

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arXiv2021

Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation

In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^\infty$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator: first, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counter-part of the Carleman representation to derive uniform control of the gain term for the relativistic collision operator. This allows us to prove the desired propagation of the uniform bounds of the solution. We further present two applications of the propagation of the uniform upper bounds: first we give another proof of the Boltzmann $H$-theorem, and second we prove the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium.

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