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The aim of the workshop is to bring together junior researchers to share their recent research findings on the mathematical analysis of various kinetic models and related systems. This event will provide a platform for exchanging ideas and fostering collaborations across diverse areas of kinetic theory, including both theoretical advancements and applications. The workshop encourages the active involvement of undergraduate and graduate students, as well as postdoctoral researchers, offering them an opportunity to present their work and receive feedback from peers and experts. Please send an email to ypchoi at yonsei.ac.kr if you need further information.
Date: September 20, 2024
Venue: Science Hall 225, Yonsei University
Registration (closed)
Confirmed Participants
Venue: Science Hall 225, Yonsei University
Registration (closed)
Confirmed Participants
Invited Speakers:
Gayoung An (Pohang University of Science and Technology)
Junhyeok Byeon (Seoul National University)
Chiara Cicolani (University of L'Aquila)
Ki Nam Hong (Sungkyunkwan University)
Dohyeon Kim (California Institute of Technology)
Jongin Kim (Pohang University of Science and Technology)
Junsung Kim (Pohang University of Science and Technology)
Dowan Koo (Yonsei University)
Sungbin Park (Pohang University of Science and Technology)
Sung-jun Son (Pohang University of Science and Technology)
Sihyun Song (Yonsei University)
Xinyu Wang (Seoul National University)
Yeongseok Yoo (Yonsei University)
Talk title/abstract and Schedule:
September 20, 2024 (09:55 - 10:00)Young-Pil Choi (Yonsei University)
Opening
Opening
September 20, 2024 (10:00 - 10:25)
Title: Stationary Boltzmann Equation for Polyatomic Gases in a slab
Abstract: In this talk, we consider the boundary value problem of the stationary polyatomic Boltzmann equation posed in one dimensional slab. More precisely, we show that under suitable norm bound assumptions on the boundary condition functions, there exists a unique mild solution to the stationary polyatomic Boltzmann equation with a convex combination of inflow and diffusive boundary conditions in a slab.
Ki Nam Hong (Sungkyunkwan University)
Title: Stationary Boltzmann Equation for Polyatomic Gases in a slab
Abstract: In this talk, we consider the boundary value problem of the stationary polyatomic Boltzmann equation posed in one dimensional slab. More precisely, we show that under suitable norm bound assumptions on the boundary condition functions, there exists a unique mild solution to the stationary polyatomic Boltzmann equation with a convex combination of inflow and diffusive boundary conditions in a slab.
September 20, 2024 (10:25 - 10:50)
Title: Global stability of the Boltzmann equation for a polyatomic gas with initial data allowing large oscillation
Abstract: In this talk, we consider the Boltzmann equation for a polyatomic gas. We establish that the mild solution to the Boltzmann equation on the torus is globally well-posed, provided the initial data satisfies bounded velocity-weighted $L^\infty$-norm and the smallness condition on the initial relative entropy. Furthermore, we also study the asymptotic behavior of solutions, converging to the global Maxwellian with an exponential rate. A key point in the proof is to develop the pointwise estimate on the gain term of non-linear collision operator for Gr\"{o}nwall's argument.
Sung-jun Son (Pohang University of Science and Technology)
Title: Global stability of the Boltzmann equation for a polyatomic gas with initial data allowing large oscillation
Abstract: In this talk, we consider the Boltzmann equation for a polyatomic gas. We establish that the mild solution to the Boltzmann equation on the torus is globally well-posed, provided the initial data satisfies bounded velocity-weighted $L^\infty$-norm and the smallness condition on the initial relative entropy. Furthermore, we also study the asymptotic behavior of solutions, converging to the global Maxwellian with an exponential rate. A key point in the proof is to develop the pointwise estimate on the gain term of non-linear collision operator for Gr\"{o}nwall's argument.
September 20, 2024 (10:50 - 11:15)
Title: On the Large amplitude solution of the Boltzmann equation with Large external potential and Boundary effects
Abstract: In this talk, we discuss the large amplitude solution of the Boltzmann equation with a large external potential \Phi(x) and the diffuse reflection boundary conditions. We first introduce L^2-L^\infty bootstrap argument to establish the asymptotic stability of small perturbations near local Maxwellian \mu_E(x,v) = e^{-\Phi(x)}e^{-\frac{|v|^2}{2}}. Subsequently, we present the asymptotic stability of large amplitude solutions with initial data that is arbitrarily large in L^{\infty}, but sufficiently small in the sense of relative entropy. This is a joint work with D. Lee (POSTECH).
Jongin Kim (Pohang University of Science and Technology)
Title: On the Large amplitude solution of the Boltzmann equation with Large external potential and Boundary effects
Abstract: In this talk, we discuss the large amplitude solution of the Boltzmann equation with a large external potential \Phi(x) and the diffuse reflection boundary conditions. We first introduce L^2-L^\infty bootstrap argument to establish the asymptotic stability of small perturbations near local Maxwellian \mu_E(x,v) = e^{-\Phi(x)}e^{-\frac{|v|^2}{2}}. Subsequently, we present the asymptotic stability of large amplitude solutions with initial data that is arbitrarily large in L^{\infty}, but sufficiently small in the sense of relative entropy. This is a joint work with D. Lee (POSTECH).
September 20, 2024 (11:15 - 11:40)
Title: High-velocity tails of the inelastic and the multi-species mixture Boltzmann equations
Abstract: We study high-velocity tails of some homogeneous Boltzmann equations. First, we consider spatially homogeneous inelastic Boltzmann equation with noncutoff collision kernel, in the case of moderately soft potentials. We also study spatially homogeneous mixture Boltzmann equations : for both noncutoff collision kernel with moderately soft potentials and cutoff collision kernel with hard potentials. In the case of noncutoff inelastic Boltzmann, we obtain \[ f(t,v) \geq a(t) e^{-b(t)|v|^p}, \quad 2 < p < 6.213 \] by extending Cancellation lemma and spreading lemma and assuming $f\in C^{\infty}$. For the Mixture type Boltzmann equations, we prove Maxwellian $p=2$.
Gayoung An (Pohang University of Science and Technology)
Title: High-velocity tails of the inelastic and the multi-species mixture Boltzmann equations
Abstract: We study high-velocity tails of some homogeneous Boltzmann equations. First, we consider spatially homogeneous inelastic Boltzmann equation with noncutoff collision kernel, in the case of moderately soft potentials. We also study spatially homogeneous mixture Boltzmann equations : for both noncutoff collision kernel with moderately soft potentials and cutoff collision kernel with hard potentials. In the case of noncutoff inelastic Boltzmann, we obtain \[ f(t,v) \geq a(t) e^{-b(t)|v|^p}, \quad 2 < p < 6.213 \] by extending Cancellation lemma and spreading lemma and assuming $f\in C^{\infty}$. For the Mixture type Boltzmann equations, we prove Maxwellian $p=2$.
September 20, 2024 (11:40 - 12:05)
Title: Exponential Type Ill-Posedness and Well-Posedness in the BGK Model and the Boltzmann Equation
Abstract: The Boltzmann equation is a fundamental equation describing the dynamics of a gas system. The BGK model is introduced to replace the complicated collision term in the Boltzmann equation for computational benefit. In this presentation, we discuss an exponential type contrast between the Boltzmann equation and the BGK equation. For this purpose, we will construct a solution in the BGK equation that blows up immediately in time in the exponential class and prove a local well-posedness theorem in the Boltzmann equation in the class. This presentation is based on the joint work with Donghyun Lee and Seok-Bae Yun.
Sungbin Park (Pohang University of Science and Technology)
Title: Exponential Type Ill-Posedness and Well-Posedness in the BGK Model and the Boltzmann Equation
Abstract: The Boltzmann equation is a fundamental equation describing the dynamics of a gas system. The BGK model is introduced to replace the complicated collision term in the Boltzmann equation for computational benefit. In this presentation, we discuss an exponential type contrast between the Boltzmann equation and the BGK equation. For this purpose, we will construct a solution in the BGK equation that blows up immediately in time in the exponential class and prove a local well-posedness theorem in the Boltzmann equation in the class. This presentation is based on the joint work with Donghyun Lee and Seok-Bae Yun.
September 20, 2024 (12:05 - 13:35)
Lunch & Coffee
Lunch & Coffee
September 20, 2024 (13:35 - 14:00)
Title: Global Existence of Mild Solutions to a BGK Model for Barotropic Gas Dynamics
Abstract: We establish global existence of mild solutions to the BGK model proposed by Bouchut under the minimal assumption of finite kinetic entropy initial data and in the full range of gamma. Moreover we rigorously derive a kinetic entropy inequality, which combined with the theory developed by Berthelin and Vasseur 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.
Sihyun Song (Yonsei University)
Title: Global Existence of Mild Solutions to a BGK Model for Barotropic Gas Dynamics
Abstract: We establish global existence of mild solutions to the BGK model proposed by Bouchut under the minimal assumption of finite kinetic entropy initial data and in the full range of gamma. Moreover we rigorously derive a kinetic entropy inequality, which combined with the theory developed by Berthelin and Vasseur 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.
September 20, 2024 (14:00 - 14:25)
Title: Which of solutions for non-homogeneous wave equation satisfies the Maxwell’s system?
Abstract: In the undergraduate electromagnetic dynamics course, it is well known that every sufficiently smooth electro-magnetic fields which satisfy the Maxwell’s system undergo the non-homogeneous wave equations, respectively where the source terms are from charge and current densities. In this talk, we prove the partial converse in that we assume that the two vector fields satisfy the wave equations, and Maxwell’s equation at initial time and charge and current densities obey the continuity equation. Here, we proved that one can recover the Maxwell’s equations in all time.
Junsung Kim (Pohang University of Science and Technology)
Title: Which of solutions for non-homogeneous wave equation satisfies the Maxwell’s system?
Abstract: In the undergraduate electromagnetic dynamics course, it is well known that every sufficiently smooth electro-magnetic fields which satisfy the Maxwell’s system undergo the non-homogeneous wave equations, respectively where the source terms are from charge and current densities. In this talk, we prove the partial converse in that we assume that the two vector fields satisfy the wave equations, and Maxwell’s equation at initial time and charge and current densities obey the continuity equation. Here, we proved that one can recover the Maxwell’s equations in all time.
September 20, 2024 (14:25 - 14:50)
Title: Clustering behavior of the kinetic Cucker-Smale model with non-compact support
Abstract: Clustering behavior is ubiquitous in natural phenomena, such as birds flocking and fish swarming. In 2007, Cucker and Smale introduced the well-known Cucker-Smale model to describe the collective behavior of bird flocks. The population of birds and fish in nature is very large, therefore how to characterize its clustering behavior when the system size goes to infinity is an important issue. The kinetic model provides such a tool to describe the clustering dynamics of the infinite particle system using the density function. In this talk, we will discuss the clustering dynamics of the kinetic Cucker-Smale model with non-compact support.
Xinyu Wang (Seoul National University)
Title: Clustering behavior of the kinetic Cucker-Smale model with non-compact support
Abstract: Clustering behavior is ubiquitous in natural phenomena, such as birds flocking and fish swarming. In 2007, Cucker and Smale introduced the well-known Cucker-Smale model to describe the collective behavior of bird flocks. The population of birds and fish in nature is very large, therefore how to characterize its clustering behavior when the system size goes to infinity is an important issue. The kinetic model provides such a tool to describe the clustering dynamics of the infinite particle system using the density function. In this talk, we will discuss the clustering dynamics of the kinetic Cucker-Smale model with non-compact support.
September 20, 2024 (14:50 - 15:15)
Title: Asymptotic dynamics of different kinds of opinion formation models with leadership and time-delayed coupling
Abstract: We present some recent results we obtained in the analysis of the convergence to consensus of different types of opinion formation models, where we considered the presence of leadership, time-delayed coupling, and non-universal interaction. The first model describes the dynamics of two groups of agents interacting with each other via subsets of individuals, namely the leaders, and natural time delay effects are considered. By using careful estimates of the system’s trajectories, we can prove an asymptotic convergence to a consensus result. This model can be generalized considering a network topology over the structure of the model. Indeed, in the second example, we deal with a Hegselmann-Krause opinion formation model with non-universal interaction, time delay, and possible lack of connection between the agents. More precisely, we analyze the situation in which the agents involved in the opinion formation do not transmit information to all the other agents of the system. Also, agents linked to each other can suspend their interaction at certain times. Under the so-called Persistence Excitation condition, we establish the exponential convergence to consensus for the Hegselmann-Krause model, whenever the digraph that describes the interaction between the agents is strongly connected. We can perform the same analysis on the second-order model, the so-called Cucker-Smale model, and we can establish the exponential asymptotic flocking for it. Finally, we consider a different kind of structure for a Hegselmann-Krause type opinion formation model in which non-universal interaction and time-delayed coupling are considered. Here, we assume a property that ensures the presence of a common influencer between two different states. Moreover, we explore two cases in which this property does not hold, where leaders with independent opinion dynamics appear. By using careful estimates of the system’s trajectories, we are able to prove an asymptotic convergence to a consensus result. We also present a possible extension of this study considering the continuum model obtained as the mean-field limit of the particle system when the size of the population tends to infinity, under an assumption on the ratio between the number of leaders and the size of the population.
Chiara Cicolani (University of L'Aquila)
Title: Asymptotic dynamics of different kinds of opinion formation models with leadership and time-delayed coupling
Abstract: We present some recent results we obtained in the analysis of the convergence to consensus of different types of opinion formation models, where we considered the presence of leadership, time-delayed coupling, and non-universal interaction. The first model describes the dynamics of two groups of agents interacting with each other via subsets of individuals, namely the leaders, and natural time delay effects are considered. By using careful estimates of the system’s trajectories, we can prove an asymptotic convergence to a consensus result. This model can be generalized considering a network topology over the structure of the model. Indeed, in the second example, we deal with a Hegselmann-Krause opinion formation model with non-universal interaction, time delay, and possible lack of connection between the agents. More precisely, we analyze the situation in which the agents involved in the opinion formation do not transmit information to all the other agents of the system. Also, agents linked to each other can suspend their interaction at certain times. Under the so-called Persistence Excitation condition, we establish the exponential convergence to consensus for the Hegselmann-Krause model, whenever the digraph that describes the interaction between the agents is strongly connected. We can perform the same analysis on the second-order model, the so-called Cucker-Smale model, and we can establish the exponential asymptotic flocking for it. Finally, we consider a different kind of structure for a Hegselmann-Krause type opinion formation model in which non-universal interaction and time-delayed coupling are considered. Here, we assume a property that ensures the presence of a common influencer between two different states. Moreover, we explore two cases in which this property does not hold, where leaders with independent opinion dynamics appear. By using careful estimates of the system’s trajectories, we are able to prove an asymptotic convergence to a consensus result. We also present a possible extension of this study considering the continuum model obtained as the mean-field limit of the particle system when the size of the population tends to infinity, under an assumption on the ratio between the number of leaders and the size of the population.
September 20, 2024 (15:15 - 15:45)
Coffee
Coffee
September 20, 2024 (15:45 - 16:10)
Title: Uniform-in-time Mean Field Limit of Consensus-Based Optimization
Abstract: Consensus-Based Optimization (CBO) algorithms are a recent family of particle methods for solving complex non-convex optimization problems. In many application settings, the objective function is not available in closed form. Additionally, derivatives may not be available, or very costly to obtain. CBO uses the Laplace principle to circumvent the use of gradients and is well-suited for black-box objectives. Most of the available analysis for this recent family of algorithms studies the corresponding mean-field descriptions of the distribution of particles. Convergence analysis with explicit rates is especially of interest in assessing algorithm performance and has mostly been done on the level of the mean-field PDEs. However, all results currently in the literature connecting the discrete particle system to the mean-field regime are restricted to finite time domains. In this talk, we present recent advances regarding the CBO algorithm and its variants and discuss uniform-in-time mean field limits. We touch on both first and second-order models. Second-order versions have numerical advantages in terms of convergence and provide a conceptual bridge to Particle Swarm Optimization (PSO), one of the most widely used particle-based optimization methods.
Dohyeon Kim (California Institute of Technology)
Title: Uniform-in-time Mean Field Limit of Consensus-Based Optimization
Abstract: Consensus-Based Optimization (CBO) algorithms are a recent family of particle methods for solving complex non-convex optimization problems. In many application settings, the objective function is not available in closed form. Additionally, derivatives may not be available, or very costly to obtain. CBO uses the Laplace principle to circumvent the use of gradients and is well-suited for black-box objectives. Most of the available analysis for this recent family of algorithms studies the corresponding mean-field descriptions of the distribution of particles. Convergence analysis with explicit rates is especially of interest in assessing algorithm performance and has mostly been done on the level of the mean-field PDEs. However, all results currently in the literature connecting the discrete particle system to the mean-field regime are restricted to finite time domains. In this talk, we present recent advances regarding the CBO algorithm and its variants and discuss uniform-in-time mean field limits. We touch on both first and second-order models. Second-order versions have numerical advantages in terms of convergence and provide a conceptual bridge to Particle Swarm Optimization (PSO), one of the most widely used particle-based optimization methods.
September 20, 2024 (16:10 - 16:35)
Title: Discrete Consensus-Based Optimization
Abstract: Recently, Consensus-Based Optimization (CBO) has been introduced as a multi-agent method for the global optimization of possibly non-convex and non-differentiable functions. CBO-type models promote consensus among agents toward a global optimum through simple stochastic dynamics, and their behaviors have generally been studied in a mean-field formulation involving a nonlinear Fokker–Planck equation. Despite the potential, a gap remains between the theoretical analysis of CBO and the actual behavior of agents in its time-discrete implementation. In this talk, we propose Discrete CBO (DCBO), which aligns with the CBO paradigm yet distinguishes itself by having no continuous counterpart. We present conditions, independent of the number of agents, for achieving consensus or convergence and study the circumstances under which global optimization occurs. We also demonstrate that DCBO is applicable to a diverse range of real-world problems with competitive performance.
Junhyeok Byeon (Seoul National University)
Title: Discrete Consensus-Based Optimization
Abstract: Recently, Consensus-Based Optimization (CBO) has been introduced as a multi-agent method for the global optimization of possibly non-convex and non-differentiable functions. CBO-type models promote consensus among agents toward a global optimum through simple stochastic dynamics, and their behaviors have generally been studied in a mean-field formulation involving a nonlinear Fokker–Planck equation. Despite the potential, a gap remains between the theoretical analysis of CBO and the actual behavior of agents in its time-discrete implementation. In this talk, we propose Discrete CBO (DCBO), which aligns with the CBO paradigm yet distinguishes itself by having no continuous counterpart. We present conditions, independent of the number of agents, for achieving consensus or convergence and study the circumstances under which global optimization occurs. We also demonstrate that DCBO is applicable to a diverse range of real-world problems with competitive performance.
September 20, 2024 (16:35 - 17:00)
Title: Global existence of weak solutions for the nonlinear Vlasov-Fokker-Planck equation
Abstract: In this talk, we discuss the global existence of weak solutions for the nonlinear Vlasov-Fokker-Planck equation. The Fokker-Planck equation is a fundamental tool for modeling how the probability distribution of particles evolves under the effects of diffusion and friction. While the existence of weak solutions for the Vlasov-Fokker-Planck equation with constant temperature is already established, there have been no results for cases with nonconstant temperature. We present the global existence of weak solutions for the nonlinear Vlasov-Fokker-Planck equation with initial data, ensuring that the conservation laws of mass, momentum, energy, and Boltzmann’s H-theorem are preserved.
Yeongseok Yoo (Yonsei University)
Title: Global existence of weak solutions for the nonlinear Vlasov-Fokker-Planck equation
Abstract: In this talk, we discuss the global existence of weak solutions for the nonlinear Vlasov-Fokker-Planck equation. The Fokker-Planck equation is a fundamental tool for modeling how the probability distribution of particles evolves under the effects of diffusion and friction. While the existence of weak solutions for the Vlasov-Fokker-Planck equation with constant temperature is already established, there have been no results for cases with nonconstant temperature. We present the global existence of weak solutions for the nonlinear Vlasov-Fokker-Planck equation with initial data, ensuring that the conservation laws of mass, momentum, energy, and Boltzmann’s H-theorem are preserved.
September 20, 2024 (17:00 - 17:25)
Dowan Koo (Yonsei University)
Title: Derivation of Euler-Poisson equations for ions
Abstract: In this talk, we establish the hydrodynamic limit of compressible Euler-Poisson equation for ions via Vlasov-Poisson equation coupled with nonlinear Fokker-Planck or certain type of BGK operators. The main tool employed in this talk is the relative entropy method. This is a joint work with Young-Pil Choi and Sihyun Song.
Dowan Koo (Yonsei University)
Title: Derivation of Euler-Poisson equations for ions
Abstract: In this talk, we establish the hydrodynamic limit of compressible Euler-Poisson equation for ions via Vlasov-Poisson equation coupled with nonlinear Fokker-Planck or certain type of BGK operators. The main tool employed in this talk is the relative entropy method. This is a joint work with Young-Pil Choi and Sihyun Song.
September 20, 2024 (17:25 - 17:30)
Byung-Hoon Hwang (Sangmyung University)
Closing
Byung-Hoon Hwang (Sangmyung University)
Closing
Organizer:
Young-Pil Choi (Yonsei University) and Byung-Hoon Hwang (Sangmyung University)
This scientific event is sponsored by Global Basic Research Lab and BK21 Yonsei Mathematical Sciences and Computation.
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