Nonuniqueness of Weak Solutions to the SQG Equation
By T. Buckmaster, Steve Shkoller, Vlad Vicol. Communications on Pure and Applied Mathematics, 2016
171 citations recorded by the source corpus.
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. © 2019 Wiley Periodicals, Inc.
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/a74da98e9315586ac87fa97f902cd19baa458f92
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In Applied Mathematics
- T. Buckmaster
- Vlad Vicol
- Camillo De Lellis
- L. Sz'ekelyhidi
- Philip Isett
- Maria Colombo
- Martina Hofmanov'a
- Antoine Choffrut
In Control and Systems Engineering
- L. Sz'ekelyhidi
- Camillo De Lellis
- T. Buckmaster
- Vlad Vicol
- Antoine Choffrut
- Jacob Bedrossian
- Nader Masmoudi
In Mathematical Physics
- T. Buckmaster
- Camillo De Lellis
- L. Sz'ekelyhidi
- Vlad Vicol
- Martina Hofmanov'a
- Maria Colombo
- Elia Brué
- Sara Daneri
Citations in the corpus
- Convex integration and phenomenologies in turbulence (2019, background): “Currently, the best known result in 2D is that there exist non-conservative weak solutions to the Euler equations lying in any Hölder space with Hölder exponent less that 1/5 [29, 28, 19, 155].”
- Nonuniqueness of weak solutions to the Navier-Stokes equation (2017, unknown): “In order to treat the dissipative term −ν∆, not present in the Euler system, we cannot proceed as in [6, 9], since in these works Hölder continuous weak solutions are constructed, which is possible only by using building blocks which are sparse in the frequency variable and for small fractional powe”
- Onsager's Conjecture for Admissible Weak Solutions (2017, background): “Indeed, in [21,22] they were able to give the first examples of, respectively, continuous and Höldercontinuous solutions that dissipate the energy, reaching the threshold exponent 1=10 (see also [7, 28] for constructions of Hölder-continuous weak solutions for other hydrodynamic models such as the I”