Caltech/UCLA/USC Joint Analysis Seminar

I will talk about joint work with A. Kiselev (Duke) on patch solutions of the Euler and alpha-SQG equations. It is well-known that the vortex patch of the 2D Euler equation is globally well-posed in non-endpoint Holder spaces. We prove that the Euler vortex patch is ill-posed at the C^2 endpoint by showing the existence of a patch with C^2 initial data such that the curvature of the patch boundary becomes infinite instantaneously. The alpha-SQG equations are a family of active scalar interpolating the 2D Euler and SQG equations. In contrast to the Euler case, we show that the alpha-SQG patch, in a suitable regime of regularity, is ill-posed in all non-L^2 Sobolev spaces and Holder spaces.