Nonuniqueness of Weak Solutions to the SQG Equation

Tristan Buckmaster, Steve Shkoller, Vlad Vicol

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)1809-1874
Number of pages66
JournalCommunications on Pure and Applied Mathematics
Volume72
Issue number9
DOIs
StatePublished - Sep 1 2019

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Nonuniqueness
Weak Solution
Dissipative Equations
Weak Compactness
Square root
Function Space
Dissipation
Open Problems
Fractional

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Nonuniqueness of Weak Solutions to the SQG Equation. / Buckmaster, Tristan; Shkoller, Steve; Vicol, Vlad.

In: Communications on Pure and Applied Mathematics, Vol. 72, No. 9, 01.09.2019, p. 1809-1874.

Research output: Contribution to journalArticle

Buckmaster, Tristan ; Shkoller, Steve ; Vicol, Vlad. / Nonuniqueness of Weak Solutions to the SQG Equation. In: Communications on Pure and Applied Mathematics. 2019 ; Vol. 72, No. 9. pp. 1809-1874.
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