Nonuniqueness of Weak Solutions to the SQG Equation

Tristan Buckmaster; Steve Shkoller; Vlad Vicol

doi:10.1002/cpa.21851

Nonuniqueness of Weak Solutions to the SQG Equation

Tristan Buckmaster, Steve Shkoller, Vlad Vicol

Mathematics

Research output: Contribution to journal › Article › peer-review

65 Scopus citations

Abstract

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 of De Lellis and Székelyhidi in 2012. 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 Marchand in 2008, 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.

Original language	English (US)
Pages (from-to)	1809-1874
Number of pages	66
Journal	Communications on Pure and Applied Mathematics
Volume	72
Issue number	9
DOIs	https://doi.org/10.1002/cpa.21851
State	Published - Sep 2019

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1002/cpa.21851

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AB - We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 of De Lellis and Székelyhidi in 2012. 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 Marchand in 2008, 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.

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Nonuniqueness of Weak Solutions to the SQG Equation

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