In the first part of this talk, I will give a brief introduction to the 2D Euler equations which describe the dynamics of 2D perfect fluids. I will present the known results on the equations with an emphasis on long-time dynamics as well as 2 key conjectures by Shnirelman and Sverak that guide the research in the field today.
In the second part of this talk, I will present recent results obtained in collaboration with Tarek M. Elgindi and Ryan M. Murray. We give a new supercritical class of data for the 2D Euler equation that allows for unbounded vorticities well beyond the Yudovich class. Within this class, we can demonstrate local existence and uniqueness of the solutions. Furthermore, we construct data for which a finite-time blow-up occurs. Leveraging the singularity formation we give an example of a finite time, infinite dimensional bifurcation for weak solutions of the 2D Euler equation in Lploc(R2) for p > 1.
