Cusp Formation in Vortex Patches: Renormalized Corner Dynamics and Low-Regularity Stability
김동하
129동 301호
0
1470
06.02 11:17
| 구분 | 편미분방정식,초청강연 |
|---|---|
| 일정 | 2026-06-11(목) 11:00~12:00 |
| 세미나실 | 129동 301호 |
| 강연자 | 조민준 (Max Planck Institute for Mathematics in the Sciences) |
| 담당교수 | 기타 |
| 기타 |
일시: 2026년 6월 11일 (목) 12:00 - 13:00
장소: 129동 301호
연사: 조민준 (Max Planck Institute for Mathematics in the Sciences)
제목: Cusp Formation in Vortex Patches: Renormalized Corner Dynamics and Low-Regularity Stability
초록: Vortex patches are weak solutions of the two-dimensional incompressible Euler equations whose vorticity is the characteristic function of a moving domain. While smooth patch boundaries are known to remain smooth globally in time, much less is understood about the evolution of singular boundary geometry. In this talk, I will discuss cusp formation from acute corners in vortex patches. The main result proves that any acute corner instantaneously develops into a cusp at the image of the corner under the flow, resolving a conjecture of Cohen–Danchin (2000). The proof reveals a renormalized dynamics governing the corner: near the singular point, the singular part of the Biot–Savart law suggests the time scale τ = t∣log r∣ and leads to a singular nonlinear ODE system for the angle and direction of the corner. We verify that the effective angle closes over long times in this renormalized variable, and then show, through a low-regularity stability argument based on the normalized symmetric difference of patches, that the true Euler patch shadows this effective dynamics. For regular corners, the estimates also yield an order-one cusp with logarithmic sharpness. This talk is based on joint work with Tarek M. Elgindi (to appear in Duke Mathematical Journal).
