Remarks on geometric properties of SQG sharp fronts and $\alpha$-patches
  Guided by numerical simulations, we present the proof of two results concerning the behaviour of SQG sharp fronts and $\alpha$-patches. We establish that ellipses are not rotational solutions and we prove that initially convex interfaces may lose this property in finite time. 
Angel Castro, Diego C\'ordoba, Javier G\'omez-Serrano, Alberto   Mart\'in Zamora
math.AP