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We give an alternative proof of the nonuniqueness of weak solutions to thesurface quasigeostrophic equation (SQG) first shown in[Buckmaster-Shkoller-Vicol, '16]. Our approach proceeds directly at the levelof the scalar field. Furthermore, we prove that every smooth scalar field withcompact support that conserves the integral can be realized as a weak limit ofsolutions to SQG.

A direct approach to nonuniqueness and failure of compactness for the SQG equation