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

2018/12/20 by Matthew Novack, Novack, Matthew
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1812.08734

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

arxiv created 2020/04/25 · arxiv updated 2020/04/28

Abstract

We show that weak solutions to the 3D quasi-geostrophic system in the class C^ζ_t,x for ζ<(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 \citecdlsj12 and recovers a result which can already be inferred from the arguments of Buckmaster, De Lellis, Isett, and Székelyhidi \citebdlisj15 or Buckmaster, Shkoller, and Vicol \citebsv16.

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