Differential Equations Seminar

We give a complete description of the long-time behavior of uniformly bounded scale-invariant solutions to the 2d Euler equation satisfying a discrete symmetry. We show that all such solutions relax in infinite time to rigidly rotating states or steady states. Consequently, all sufficiently symmetric non-constant scale-invariant solutions that are smooth on S^1 become singular in infinite time. On the plane, this corresponds to generic infinite time spiral and cusp formation for bounded and discretely symmetric solutions. This is based on joint works with R. Murray and A. Said. Speaker(s): Tarek Elgindi (Duke University)