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Nonuniqueness of weak solutions to the SQG equation. (English) Zbl 1427.35200

Summary: We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 of C. De Lellis and L. Székelyhidi jun. [Bull. Am. Math. Soc., New Ser. 49, No. 3, 347–375 (2012; Zbl 1254.35180)]. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. In view of the results of F. Marchand [Commun. Math. Phys. 277, No. 1, 45–67 (2008; Zbl 1134.35025)], we establish that for the dissipative SQG equation, weak solutions may be constructed in the same function space both via classical weak compactness arguments and via convex integration.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q86 PDEs in connection with geophysics
76W05 Magnetohydrodynamics and electrohydrodynamics
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
35R11 Fractional partial differential equations
35D30 Weak solutions to PDEs
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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