Nonuniqueness of Weak Solutions to the 3 Dimensional Quasi-Geostrophic Equations

Abstract

We show that weak solutions to the 3 dimensional quasi-geostrophic system in the class $C^\zeta_{t,x}$ for $\zeta<\frac{1}{5}$ are not unique and may achieve any smooth, nonnegative energy profile. Our proof follows a convex integration scheme which utilizes in a crucial way the stratified nature of the quasi-geostrophic velocity field, providing a link with the 2 dimensional (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 Szekélyhidi [preprint, arxiv.org/abs/1205.1226, 2012] and recovers a result which can already be inferred from the arguments of Buckmaster et al. [Ann. of Math. (2), 182 (2015), pp. 127--172], or Buckmaster, Shkoller, and Vicol [Comm. Pure Appl. Math., 12 (2019), pp. 1809--1874].