Non-uniqueness of Weak Solutions to the 3D Quasi-Geostrophic Equations

We show that weak solutions to the 3D quasi-geostrophic system in the class $C^\zeta_{t,x}$ for $\zeta<\frac{1}{5}$ are not unique and may achieve any smooth, non-negative energy profile. Our proof follows a convex integration scheme which utilizes the stratified nature of the quasi-geostrophic velocity field, providing a link with the 2D Euler equations. In fact we observe that under particular circumstances our construction coincides with the convex integration scheme for the 2D Euler equations introduced by Choffrut, De Lellis, and Szek\'{e}lyhidi \cite{cdlsj12} and recovers a result which can already be inferred from the arguments of Buckmaster, De Lellis, Isett, and Sz\'{e}kelyhidi \cite{bdlisj15} or Buckmaster, Shkoller, and Vicol \cite{bsv16}.