Singularity formation in the contour dynamics for 2d Euler equation on the plane

We will study 2d Euler dynamics of centrally symmetric pair of patches on the plane. In the presence of exterior regular velocity, we will show that these patches can merge so fast that the distance between them allows double-exponential upper bound which is known to be sharp. The formation of the 90 degree corners on the interface and the self-similarity analysis of this process will be discussed. For a model equation, we will discuss existence of the curve of smooth stationary solutions that originates at singular stationary steady state.