Analysis, PDE & Probability Seminar

Upcoming Seminars

2026. 7. 8 (Wed) 15:30-18:00 Room 1424
Changfeng Gui (University of Macau)

Classifications of steady solutions to two-dimensional Euler equations

2026. 7. 14 (Tue) 16:00-17:00 Room 7323
Kiyeon Lee (KAIST)

TBA

2026. 7. 15 (Wed) 16:00-17:00 Room 7323
Gichan Bae (Seoul National University)

Quantitative Closure Analysis toward Ideal Fluids

2026. 7. 16 (Thu) 15:30-17:30 Room 1424
Jaeheang Bang (Westlake University)

Sharp Ill-Posedness of the Euler Equations in Lorentz Spaces

ABSTRACT. We are concerned about vortex stretching for the three-dimensional axisymmetric Euler equations without swirl in vorticity formulation. Danchin (2007) established global existence and uniqueness for bounded vorticity $\omega_0$ provided the relative vorticity $\omega_0/r$ lies in the endpoint Lorentz space $L^{3,1}(\mathbb{R}^3)$ (together with a decay assumption on $\omega_0$). We proved that this $L^{3,1}$ endpoint is sharp: for every Lorentz exponent $q>1$, we construct multi-ring data $\omega_0 \in L^\infty (\mathbb{R}^3)$ with $\omega_0/r\in L^{3,q}(\mathbb{R}^3)$ that produce $L^\infty$-norm inflation of the vorticity; moreover, within the same class, we obtain instantaneous blow-up from data with infinitely many rings. Our initial data are inspired by the Kim--Jeong dyadic ring superposition (2022), but we crucially generalize it by allowing flexible conical support geometry for the ring profile. In the regime where outer rings are dominant---a multiscale viewpoint appearing in recent works including Kim--Jeong (2022) and Cordoba--Martinez-Zoroa--Zheng (2025)---we obtain a forward-in-time ODE cascade for ring amplitudes and aspect ratios in which vortex stretching weakens its own future forcing: as a ring amplifies, incompressibility flattens it, the aspect ratio collapses, and the induced stretching coefficient is geometrically depleted. A key new ingredient is a profile-localization argument that freezes the relevant Biot–Savart kernel and makes this depletion explicit, enabling us to exploit a monotone ``productive window" (controlled by the cone slope) together with an exact cascade identity. This propagates stretching across scales and gives a robust lower bound on cumulative stretching, yielding ill-posedness in the full range $q>1$. If time permits, I will also describe how to transfer norm inflation to instantaneous blow up and main difficulties of the study of our limiting distributional solutions. This is a joint work with Alexey Cheskidov (Westlake University).

2026. 7. 22 (Wed) 16:00-17:00 Room 1423
Jakwang Kim (Chinese University of Hong Kong, Shenzhen)

TBA

2026. 8. 3 (Mon) 15:00-16:00 Room 1423
Sewook Oh (GIST)

TBA

2026. 8. 3 (Mon) 16:00-17:00 Room 1423
Juyoung Lee (KIAS)

TBA

2026. 8. 4 (Tue) 15:00-16:00 Room 1424
Haesung Lee (국립금오공과대학교)

TBA

2026. 8. 4 (Tue) 16:00-17:00 Room 1424
Minjung Gim (National Institute for Mathematical Sciences)

TBA

2026. 8. 5 (Wed) 16:00-17:30 Room 1423
Soobin Cho (University of Illinois Urbana-Champaign)

TBA

2026. 8. 7 (Fri) 16:00-17:30 Room 1423
Hantaek Bae (UNIST)

TBA

2026. 8. 27 (Thu) 16:00-17:30 Room 1424
Jae-Hwan Choi (KIAS)

TBA