  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.
