##### Department of Mathematics,

University of California San Diego

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### Math 248 - Analysis Seminar

## Tarek Elgindi

#### Princeton University

## On Singular Vortex Patches

##### Abstract:

Since the seminal work of Yudovich in 1963, it has been known that for a given uniformly bounded and compactly supported initial vorticity profile, there exists a unique global solution to the 2d incompressible Euler equation. A special class of Yudovich solutions are so-called vortex patch solutions where the vorticity profile is the characteristic function of an (evolving) bounded set in $\mathbb{R}^2.$ In 1993 Chemin and Bertozzi-Constantin proved that sufficiently high regularity of the boundary is propagated for all time. Since then, there have been numerous numerical and rigorous works on understanding the long-time dynamics of smooth vortex patches as well as the short time dynamics of vortex patches with corners. In this work, we consider two regimes; one where we prove well-posedness and the other where we prove ill-posedness. First, for vortex patches with corners enjoying a certain symmetry property at the corners, we prove global propagation of the corners; we also give examples where these vortex patches cusp in infinite time. Second, we prove that vortex patches with a single corner (which do not satisfy the symmetry condition) immediately cease to have a corner. This is joint work with I. Jeong.

Host: Ioan Bejenaru

### April 18, 2017

### 10:00 AM

### AP&M 7321

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