## PDE Real Analysis Seminar

Date, time & place Tuesday 10:30 - 11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Next seminar

### 2018/12/18

10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
In-Jee Jeong (Korea Institute for Advanced Study (KIAS))
Dynamics of singular vortex patches (English)
[ Abstract ]
Vortex patches are solutions to the 2D Euler equations that are given by the characteristic function of a bounded domain that moves with time. It is well-known that if initially the boundary of the domain is smooth, the boundary remains smooth for all time. On the other hand, we consider patches with corner singularities. It turns out that, depending on whether the initial patch satisfies an appropriate rotational symmetry condition or not, the corner structure may propagate for all time or lost immediately. In the rotationally symmetric case, we are able to construct patches with interesting dynamical behavior as time goes to infinity. When the symmetry is absent, we present a simple yet formal evolution equation which describes the dynamics of the boundary. It suggests that the angle cusps instantaneously for $t > 0$.
This is joint work with Tarek Elgindi.