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

Abstract

We show that weak solutions to the 3D quasi-geostrophic system in the class $C_{t,x}^{\zeta }$ 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 Szekélyhidi \cite{cdlsj12} and recovers a result which can already be inferred from the arguments of Buckmaster, De Lellis, Isett, and Székelyhidi \cite{bdlisj15} or Buckmaster, Shkoller, and Vicol \cite{bsv16}.

arXiv comment43 pages. Added proof of non-uniqueness from arbitrary smooth inital data, minor technical issues and typos resolved. Accepted for publication in SIMA

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Cited by in corpus (6)

• Sharp nonuniqueness for the Navier-Stokes equations
• Classical Solutions for the 3D Quasi-Geostrophic System on a Bounded Domain
• The Inviscid 3D Quasi-Geostrophic System on Bounded Domains
• Nonuniqueness of weak solutions for the transport equation at critical space regularity
• Anomalous dissipation, anomalous work, and energy balance for smooth solutions of the Navier-Stokes equations
• Hölder continuous weak solution of 2d Boussinesq equation with diffusive temperature