The 12th Korea PDE Winter School

Invited Lecture Series

He Ling-Bing (Tsinghua University)

Mathematical Theory of Collisional Kinetic Equations: Derivation, Well-Posedness, and Numerical Simulation

Abstract: In this course, we provide a concise introduction to the mathematical foundations of collisional kinetic equations—specifically, the Boltzmann and Landau equations, which model particle dynamics in dilute and weak‑coupling regimes, respectively. We begin with a brief overview of their physical derivation. The core of the course covers essential analytical techniques, including changes of variables, the cancellation lemma, and sharp estimates for collision operators. We then discuss well‑posedness theory, emphasizing moment propagation, energy‑entropy methods, and perturbation frameworks—along with their role in designing and analyzing numerical simulations. The course concludes with a survey of extended models, such as kinetic equations with external potentials, the Vlasov‑Poisson and Vlasov‑Maxwell systems, and hydrodynamic limits leading to macroscopic fluid equations.

Invited Lectures

Kyudong Choi (UNIST) 

Lecture 1 An Introduction to the Two-Dimensional Incompressible Euler Equations 

Abstract: This series of lectures offers a graduate-level introduction to the two-dimensional incompressible Euler equations, with a focus on flows posed on the half-plane. Starting from the velocity formulation, we carefully derive the vorticity equation and explain how the two-dimensional setting leads to an active scalar model, emphasizing intuition and key ideas rather than technical details. We review fundamental structural properties, including conserved quantities, the flow map, and the Yudovich theory for weak solutions with bounded vorticity. As a simplest motivating example, we explain why a disk vortex patch yields a steady solution and outline the basic mechanisms underlying its stability. The lecture concludes with a discussion of the half-plane geometry via symmetry arguments and illustrative point vortex examples. 

Lecture 2 Traveling Waves and the Lamb Dipole in the Two-Dimensional Euler Equations 

Abstract: This lecture is devoted to traveling wave solutions of the two-dimensional Euler equations. We begin with dipole configurations of point vortices as a motivating example and discuss conditions under which a traveling vortex with bounded vorticity may arise. The second part of the lecture focuses on the Lamb dipole as a canonical and explicitly computable traveling wave solution. We review its explicit formulation and historical background, present both a derivation and a direct verification, and explain its characterization as an energy maximizer under suitable constraints. The lecture concludes with a discussion of uniqueness via a moving plane argument applied to an axisymmetric extension of the half-plane formulation in four dimensions. 

In-Jee Jeong (Seoul National University)

Lecture 3 Stability and instability of the Lamb dipole 

Abstract: After reviewing conserved quantities for Euler and energy maximizing property of the Lamb dipole, we explain how nonlinear orbital stability of the Lamb dipole follows from the variational principle. Stability is orbital in the sense that for each moment of time, there is some (unknown) shift of the Lamb dipole such that the solution is close to. We show how to obtain an estimate for this shift function. As an application of this shift estimate, we prove occurrence of filamentation for perturbations of the Lamb dipole, which is a particular form of nonlinear instability for the Lamb dipole. 

Lecture 4 Stability of multiple Lamb dipoles 

Abstract: If one considers a linear superposition of Lamb dipoles (or more general dipoles), even if the dipoles are very far away from each other, it cannot be a kinetic energy maximizer, even locally in the phase space. We explain recent progress in this direction, which proves that if faster Lamb dipoles are positioned to the right of slower ones and they are initially sufficiently far, such a configuration remains orbital stable for all positive times. The idea is to combine a variational approach with dynamical bootstrapping schemes, which among other things allows one to study how much energy is exchanged between the dipoles. We conclude with several open problems related to dynamics of strongly interacting dipoles.

Invited Talks

Gi-Chan Bae (Seoul National University) 

Quantitative Closure Analysis of Incompressible Hydrodynamic Limits

Abstract: The hydrodynamic limit leading to the incompressible Euler equation, without assuming prior knowledge of the limiting solution, has long been a central open problem in the theory of the Boltzmann equation. In this talk, we show that solutions of the scaled Boltzmann equation converge to solutions of the incompressible Euler equation as the Knudsen number tends to zero. The convergence holds for arbitrarily long times in two dimensions and locally in time in three dimensions. Our results hold not only when the vorticity belongs to $L^\infty_x$, but also when it lies in $L^p_x$ for $p<\infty$, or even when it is a Radon measure with a distinguished sign.

Sangdon Jin (Chungbuk National University)

On the partial semi-classical Weyl's law

Abstract: We study the partial dimensional semi-classical Weyl’s laws, describing the quantum subband structures for two-dimensional electron gases (2DEGs). As a simple application, we derive lowest free energy states for the subband models describing non-interacting 2DEGs.

Min Jun Jo (Max Planck Institute for Mathematics in the Sciences)

Cusp Formation in Vortex Patches

Abstract: Cohen and Danchin (2000) conjectured that any initial vortex patch with acute corners would instantaneously develop a cusp structure of order one with logarithmic sharpness. In this talk, based on joint work with T. Elgindi, we present a rigorous proof of this conjecture. Our key observation is that the dynamics near singular points is governed by a nonlinear singular fourth-order ODE, which elucidates the mechanism of cusp formation.

Jinwook Jung (Hanyang University)

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

Abstract: In this talk, we develop a unified relative entropy framework for macroscopic limits of kinetic equations with Riesz-type interactions and Fokker--Planck relaxation. The method combines entropy dissipation, Fisher-information control, and modulated interaction energies into a robust stability theory that yields both strong and weak convergence results. For the strong convergence, we establish quantitative relative entropy estimates toward macroscopic limits under well-prepared data, extending the scope of the method to settings where nonlocal forces and singular scalings play a decisive role. For the weak convergence, we prove that quantitative convergence propagates in bounded Lipschitz topologies, even when the initial relative entropy diverges with respect to the singular scaling parameter. This dual perspective shows that relative entropy provides not only a tool for strong convergence, but also a new mechanism to handle mildly prepared initial states. We establish quantitative convergence toward three prototypical limits: 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 analysis highlights the unifying role of relative entropy in connecting microscopic dissipation with both strong and weak macroscopic convergence. This talk is based on the joint work with Y.-P. Choi (Yonsei Univ.).

Junha Kim (Ajou University)

On wellposedness of $\alpha$-SQG equations in the half-plane 

Abstract: We investigate the wellposedness of $\alpha$-surface quasi-geostrophic ($\alpha$-SQG) equations in the half-plane, where $\alpha = 0$ and $\alpha = 1$ correspond to the 2D Euler and SQG equations respectively. For $0 < \alpha \leq 1/2$, we prove local wellposedness in certain weighted anisotropic Hölder spaces. We also show that such a wellposedness result is sharp: for any $0 < \alpha \leq 1$, we prove nonexistence of Hölder regular solutions (with the Hölder regularity depending on $\alpha$) for initial data smooth up to the boundary. This is a joint work with In-Jee Jeong(SNU) and Yao Yao(NUS).

Minhyun Kim (Hanyang University)

Green function estimates for nonlocal equations

Abstract: We establish the optimal regularity for solutions to nonlocal elliptic equations with H\"older continuous coefficients in divergence form in bounded $C^{1,\alpha}$ domains. Our proof is based on a delicate higher order Campanato-type iteration at the boundary, which we develop in the context of nonlocal equations and which is quite different from the local theory. As an application of our results, we establish sharp two-sided Green function estimates for the same class of operators.

Gyounghun Ko (Chinese Academy of Sciences) 

Global Well-posedness for the Multi-species Boltzmann Equation with Large Amplitude Initial Data

Abstract: This paper establishes the global well-posedness of the multi-species Boltzmann equation with large-amplitude initial data in the periodic domain. In contrast to the single-species case, the multi-species mixture model lacks structural symmetry in its collision operators due to the distinct masses of different species. This asymmetry makes it difficult to obtain pointwise estimates for the nonlinear collision terms. To overcome this, we have further developed the Carleman representation specifically for the mixture model. By applying this refined approach, we derive the desired pointwise estimates for the nonlinear terms. In this talk, we will focus on this developed Carleman representation and discuss how the large-amplitude solutions exist globally and exhibit exponential decay toward equilibrium when the initial relative entropy is sufficiently small.

Sang-Hyuck Moon (Pusan National University)

Geometry effects on the boundary-layer profiles of a chemotaxis model

Abstract: We consider the boundary-layer problem of a nonlocal semilinear elliptic equation in a bounded smooth domain of all dimensions with the Dirichlet boundary condition, which arises as the stationary problem describing the boundary-layer formation driven by chemotaxis. Using the Fermi coordinates and delicate analysis with subtle estimates, we rigorously derive the asymptotic expansion of the boundary-layer profile and thickness in terms of the small diffusion rate with coefficients explicitly expressed by the domain geometric properties including mean curvature, volume and surface area. By these expansions, one can explicitly find the joint impact of the mean curvature, surface area and volume of the spatial domain on the boundary-layer steepness and thickness.

January 19 (Monday), 15:10-15:40

이재용 (중앙대학교) 

Deep Learning Approaches for Solving Kinetic Equations

Abstract: Recent advances in deep learning have introduced powerful tools for solving kinetic equations, which often suffer from high computational costs due to nonlinearity and dimensionality. In this talk, we present deep learning-based approaches to solve two important kinetic models: the Vlasov–Poisson–Fokker–Planck equation and the Fokker–Planck–Landau equation. We also introduce FourierSpecNet, a neural network framework that incorporates spectral structure to efficiently approximate the Boltzmann collision operator. The method enables resolution-invariant learning and fast inference.

January 19 (Monday), 15:40-16:10 

김종인 (POSTECH)
Global solutions in L^p_v L^\infty_x​ for the Boltzmann equation in bounded domains

Abstract: The Boltzmann Equation is a fundamental kinetic equation that describes the dynamical behavior of a rarefied gas by characterizing the distribution of particle positions and velocities over time. The existence theory for solutions to the Boltzmann equation in bounded domains has primarily been developed within uniformly bounded function classes, such as L^\infty_{x,v}​. We consider solutions in relaxed function spaces L^p_v L^\infty_x for the initial-boundary value problem of the Boltzmann equation in bounded domains. We deal with the case of hard potential under diffuse reflection boundary conditions and assume cutoff model. For large initial data in a weighted L^p_v L^\infty_x space with small relative entropy, we construct unique global-in-time mild solution that converge exponentially to the global Maxwellian. This is a joint work with Dingqun Deng and Donghyun Lee.

January 20 (Tuesday), 15:10-15:40

박성빈 (POSTECH) 

Well-Posedness Theory and Lower and Upper Bounds for the Boltzmann Equation with Fermi–Dirac Statistics

Abstract: The Boltzmann-Fermi-Dirac equation, proposed by L.W. Nordheim (1928) and by E. A. Uehling and George Uhlenbeck(1933), is a quantum modification of the classical Boltzmann equation for Fermi-Dirac statistics. In this talk, we extend some results in the classical Boltzmann equation to the Boltzmann-Fermi-Dirac equation. (1) We first state the existence and uniqueness of the solution of the Boltzmann-Fermi-Dirac equation. (2) We construct a Gaussian lower bound for the solution. (3) Finally, we consider polynomial and exponential moments in the $L^1$ and $L^\infty$ settings. We will discuss the main difficulties and techniques in proving these results compared to the classical equation. This is a joint work with Gayoung An.

January 20 (Tuesday), 15:40-16:10

송시현 (연세대학교) 

Strong compactness phenomena in nonlinear and degenerate kinetic Fokker--Planck equations

Abstract: The nonlinear kinetic Fokker--Planck equation is a popular model, especially in the physics literature, owing to its analogue with the Landau and Boltzmann equations. Nevertheless its analysis is extremely difficult owing to its nonlinearity, and especially its degeneracy, that is present in the diffusion. In particular, it has remained an open problem whether a weak solution can be constructed assuming \textit{only} the conditions of finite mass, entropy, and energy on the initial data. This work resolves that problem by proving a new compactness property for the nonlinear Fokker--Planck operator. Put roughly, we show that if a sequence of functions satisfies a weighted Fisher information bound (in the $v$-variable), then this can be combined with the velocity averaging lemma (in the $t,x$-variables) to yield strong compactness in all three variables. This new compactness lemma does not depend at all on the 

January 21 (Wednesday), 15:10-15:40

심영진 (UNIST) 

 A rigorous analysis of vortex atmosphere in 2D and 3D: particles carried forward by a vortex

Abstract: In incompressible and inviscid fluids, the vortex atmosphere refers to the collection of fluid particles outside the support of a traveling vortex that are nevertheless carried along by the support. This phenomenon has been noted since the nineteenth century (byW. Thomson, L. Prandtl, J. C. Maxwell, O. Reynolds, E. F. Northrup, etc.), yet formal mathematical definitions and proofs have remained absent, with existing studies relying primarily on thin-core vortex approximations or numerical computations. In this talk, we give a rigorous definition of a vortex atmosphere and establish its existence and uniqueness. We further compare the planar atmosphere surrounding a traveling two-dimensional vortex dipole with the axisymmetric atmosphere encircling a traveling three-dimensional vortex ring. In particular, we emphasize and prove the topological distinctions: under natural assumptions, the two-dimensional vortex atmosphere together with its core admits only an oval-shaped configuration, whereas in three dimensions, both spherical and toroidal configurations can occur. The proof is obtained by showing that each atmosphere is precisely characterized as a specific level set of its corresponding stream function, thereby confirming earlier observations made by W. M. Hicks [Lond. Edinb.Dubl. Phil. Mag., 1919].

January 21 (Wednesday), 15:40-16:10

은남현 (KAIST) 

Uniqueness and stability of Riemann shocks to the full Euler system in a class of inviscid limits 

Abstract: In this talk, we will discuss the uniqueness and stability of Riemann shocks to the compressible Euler system in a physical class of vanishing dissipation limits. The Riemann shock is a fundamental example of a singular solution that arises in the compressible Euler system. It is a time-irreversible and discontinuous solution, with a self-similar structure. We focus on the one-dimensional full Euler system and consider the Brenner-Navier-Stokes-Fourier system, proposed as a modification of the Navier-Stokes-Fourier system, to describe the physical perturbation class. The proof relies on the method of a-contraction with shifts, and we will also comment on future directions towards small BV solutions. This is a joint work with Moon-Jin Kang (KAIST)and Saehoon Eo (Stanford University).

January 22 (Thursday), 15:10-15:40 

이승재 (서울대학교) 

Sharp local well-posedness of C¹ vortex patches

Abstract: It is well known that the boundary dynamics of vortex patches is globally well-posed in the Hölderspace C^{1,\alpha} for 0<\alpha<1, whereas the well-posedness in C^1 remains an open problem, even locally. In thistalk, we establish the local well-posedness for vortex patches in the space C^{1,\varphi} defined via a modulus ofcontinuity \varphi that satisfies certain structural assumptions. Our class includes curves that are strictly rougher thanthe Hölder-continuous ones, with prototypical examples being \varphi(r) = (-\log r)^{-s} for s>3. Motivated by the fact that the velocity operator in the contour dynamics equation is a nonlinear variant of the Hilbert transform, we study the system of equations satisfied by the curve parametrization \gamma \in C^{1,\varphi} and its Hilbert transform. In doing so, we derive several properties of the Hilbert transform and its variants in critical spaces, which are essential for controlling the velocity operator and its Hilbert transform.

January 22 (Thursday), 15:40-16:10 

구도완 (Oxford University)

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

In this talk, I will introduce the hydrodynamic Euler–Alignment model, focusing on the pressureless case coupled with nonlocal interaction forces, and discuss its large-time dynamics—namely, the emergence of flocking and the characterization of its asymptotic behavior. New flocking estimates will be presented, showing how the confining effect of nonlocal interaction can, in certain regimes, replace the role of velocity alignment. The quantitative analysis of the asymptotic behavior will also be discussed. Overall, the convergence rate depends only on the local behavior of the communication weight: bounded kernels lead to exponential decay, while weakly singular ones yield algebraic rates. This reveals a sharp transition in decay rates driven solely by the local singularity of the communication kernel, a regime that had remained largely unexplored. This talk is based on joint work with José Carrillo (University of Oxford), Young-Pil Choi (Yonsei University), and Oliver Tse (Eindhoven University of Technology).