Analysis, PDE & Probability Seminar

2026 Seminars

2026. 3. 10 (Tue) 11:00-12:00  Room 1423
Jaeyun Yi (KIAS HCMC)

Advection-Diffusion equation driven by random velocity

ABSTRACT. We discuss the long-time behavior of advection-diffusion equations driven by random velocity. It is well-known that the $L_2$ norm of the solution decays at least exponentially fast when the velocity field is incompressible. We know that this exponential decay is optimal in the case of pure diffusion, but is less clear once the advection is added. Establishing lower bounds on the decay rate, especially faster than exponential, has attracted significant attention in both the deterministic and stochastic PDE communities. We show that when the advection-diffusion equation is driven by a “typical random” velocity, the exponential decay is optimal. Our result partially answers the Bachelor conjecture concerning the natural length scale of passive scalars. This is joint work with Martin Hairer, Tommaso Rosati, and Sam-Punshon Smith.

2026. 3. 10 (Tue) 10:00-11:00  Room 1423
Young-Jin Sim (KIAS)

Existence and stability of Sadovskii vortex patch: an odd-symmetric touching pair of uniform vortices

ABSTRACT. As a traveling solution of the two-dimensional incompressible Euler equations, the Sadovskii vortex patch takes the form of a counter-rotating pair of vortex patches that are in contact. This model was suggested by Sadovskii [J. Appl. Math. Mech., 1971] and has since attracted considerable attention due to its significance to the inviscid limit of planar flows via the Prandtl-Batchelor theory and its role as an asymptotic state of the dynamics of vortex rings. In this paper, we establish the existence and stability of the Sadovskii vortex patch. First, we show that a Sadovskii vortex patch arises as a maximizer of the kinetic energy under an exact impulse condition. By analyzing the fluid velocity along the symmetry axis and its relation to the vorticity of a dipole, we verify that two patches in the dipole are in contact. Second, we construct a subcollection of such energy maximizers that is stable in the following sense: if an initial vorticity is sufficiently close to a maximizer, then the corresponding solution remains close to this collection up to a translation and travels at a similar speed to the maximizers. This is achieved via a concentration-compactness argument together with the conservation of impulse, circulation, and kinetic energy in time. Furthermore, using uniform estimates of energy maximizers and a shift estimate obtained by estimating the center of mass of the solution, we show that the solution keeps its touching structure uniformly in time.

2026. 3. 4 (Wed) 16:00-18:00  Room 1424
Ken Abe (Osaka Metropolitan University)

MHS equilibria in the non-resistive limit to the randomly forced resistive magnetic relaxation equations

ABSTRACT. I will discuss the question of the existence of 3D steady Euler flows (MHS equilibria) via magnetic relaxation equations, and introduce our recent study on their statistical model. This talk is based on joint work with I.-J. Jeong (KIAS), F. Pasqualotto (UCSD), and N. Sato (NIFS).

2026. 2. 24 (Tue) 16:00-17:00  Room 8309
Kenta Nakamura (Nihon University)

Volume concentration phenomenon for the p-Sobolev flow

ABSTRACT. We provide a new quantitative local boundedness for the p-Sobolev flow, which is a nonlinear generalization of the gradient flow concerning the p-Dirichlet energy on a bounded domain. In the case p=2, the p-Sobolev flow is the heat flow. This estimate enables us to observe a volume concentration phenomenon for the p-Sobolev flow. This is based on a joint work with Tuomo Kuusi and Masashi Misawa.

2026. 2. 23 (Mon) 16:00-17:00  Room 1423
Hyungsung Yun (Changwon National University)

Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations

ABSTRACT. In this talk, we study the boundary regularity of viscosity solutions to fully nonlinear degenerate or singular parabolic equations of the form $$u_t = |Du|^\gamma F(D^2u)+f, \quad \gamma>-1.$$ The gradient-dependent degeneracy or singularity, together with the time derivative, poses significant difficulties compared to the elliptic case. We establish boundary $C^{1,\alpha}$ estimates for the Dirichlet problem under suitable structural assumptions on the operator and boundary data. The proof combines compactness arguments, barrier constructions, regularization techniques, and the boundary regularity of small perturbation solutions, together with the analysis of a model problem on a flat boundary. Our results unify and extend previous boundary regularity results and also provide a framework for studying boundary behavior in a broader class of nonlinear parabolic equations.

2026. 2. 9 (Mon) 14:00-15:30  Room 1423
Jinyeop Lee (University of Basel)

The Semiclassical Limit of the 2D Dirac–Hartree Equation with Periodic Potentials

ABSTRACT. We study the semiclassical limit of the two-dimensional Dirac–Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac-Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov-Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter. This is joint work with Kunlun Qi.

2026. 2. 3 (Tue) 10:00-11:00  Room Online
Hyunwoo Kwon (Brown University)

표면장력효과를 고려한 단상 무스캣 문제의 대역해의 존재성

ABSTRACT. 단상 무스캣 문제는 다공성 매질에서 건조한 영역과 Darcy's flow에 의해 기술되는 유체가 있는 영역 사이의 경계면의 발전을 다룹니다. 표면장력효과를 고려하는 것은 물리적인 이유로도 자연스럽지만, 이 효과를 고려할 경우, 해 자체에 대해 극대원리와 같은 수단을 쓸 수가 없어 문제해결 난이도가 복잡해집니다. 이 발표에서는 전공간에서 정의되었거나 주기성을 고려해도 초기조건이 적절히 작다면, 대역해의 존재성을 보입니다. 더불어 주기성을 고려할 경우, 해의 점근성도 보입니다. 본 발표는 董弘桀(동홍지에) 교수님과의 공동연구를 바탕으로 하며, 발표자료도 한국어로 진행합니다.

2026. 1. 28 (Wed) 16:00-17:00  Room 1423
Junkee Jeon (Kyunghee University)

Stochastic Control and Free Boundary Problems Arising in Financial Mathematics

ABSTRACT. In this presentation, I discuss stochastic control problems arising from financial mathematics. Specifically, I address problems that combine stochastic control with optimal stopping, as well as those involving singular control and switching control. Furthermore, I investigate a two-person zero-sum game played between a singular controller and an optimal stopper. A key focus of this talk is deriving the associated parabolic variational inequalities and analyzing the analytic properties of the free boundaries they induce. I will also explain how these boundaries characterize the optimal strategies. Finally, I illustrate the application of these models in finance with numerical examples.

2026. 1. 14 (Wed) 16:00-17:00  Room 1423
Kihoon Seong (Cornell University)

Sine–Gordon measures, topological soliton, and Gaussian multiplicative chaos

ABSTRACT. I will discuss the sine–Gordon QFT, which lies in the same universality class as the Coulomb gas and the XY model, and exhibits the BKT phase transition (Nobel prize 2016). The sine–Gordon action admits infinitely many homotopy classes. Even though the action has no energy minimizers in homotopy classes with high topological charge, I will show that the Gibbs measure nevertheless concentrates and exhibits Ornstein–Uhlenbeck fluctuations around the multi-soliton manifold. I will then discuss the geometry of this multi-soliton manifold, including how solitons are arranged, such as their expected locations and gaps. Furthermore, I will explain the asymptotics of the partition function using Gaussian multiplicative chaos. Based on joint work with Hao Shen and Philippe Sosoe.

Past Seminars

2025 Seminars
2024 Seminars
2023 Seminars
2022 Seminars
2021 Seminars
2020 Seminars
2019 Seminars

Organizers

Jae-Hwan Choi   -   Korea Institute for Advanced Study

Namhyun Eun   -   Korea Institute for Advanced Study

In-Jee Jeong   -   Korea Institute for Advanced Study

Yong-Gwan Ji   -   Korea Institute for Advanced Study

Nam-Gyu Kang   -   Korea Institute for Advanced Study

Taegyu Kim   -   Korea Institute for Advanced Study

Wontae Kim   -   Korea Institute for Advanced Study

Joonhyun La   -   Korea Institute for Advanced Study

Jung-Kyoung Lee   -   Korea Institute for Advanced Study

Juyoung Lee   -   Korea Institute for Advanced Study

Se-Chan Lee   -   Korea Institute for Advanced Study

Sewook Oh   -   Korea Institute for Advanced Study

Sung-Jin Oh   -   UC Berkeley & KIAS

Hyangdong Park   -   Korea Institute for Advanced Study

Junhee Ryu   -   Korea Institute for Advanced Study

Jinsol Seo   -   Korea Institute for Advanced Study

Young-Jin Sim   -   Korea Institute for Advanced Study

Kyeong Song   -   Korea Institute for Advanced Study

Jaeyun Yi   -   Korea Institute for Advanced Study

Directions

Korea Institute for Advanced Study
Buildings 1 & 8, 85 Hoegi-ro, Dongdaemun-gu, Seoul, Korea



June E Huh Center for Mathematical Challenges
2nd Floor, 118 Hongneung-ro, Dongdaemun-gu, Seoul, Korea