Nonuniqueness of weak solutions to the SQG equation
  We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 in the survey arXiv:1111.2700 by De Lellis and Sz\'ekelyhidi Jr. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. 
Tristan Buckmaster, Steve Shkoller and Vlad Vicol
math.AP