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Convex integration constructions in hydrodynamics


Authors: Tristan Buckmaster and Vlad Vicol
Journal: Bull. Amer. Math. Soc. 58 (2021), 1-44
MSC (2020): Primary 35Q35
DOI: https://doi.org/10.1090/bull/1713
Published electronically: November 2, 2020
MathSciNet review: 4188806
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Abstract: We review recent developments in the field of mathematical fluid dynamics which utilize techniques that go under the umbrella name convex integration. In the hydrodynamical context, these methods produce paradoxical solutions to the fluid equations which defy physical laws. These counterintuitive solutions have a number of properties that resemble predictions made by phenomenological theories of fluid turbulence. The goal of this review is to highlight some of these similarities while maintaining an emphasis on rigorous mathematical statements. We focus our attention on the construction of weak solutions for the incompressible Euler, Navier–Stokes, and magneto-hydrodynamic equations which violate these systems’ physical energy laws.


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Additional Information

Tristan Buckmaster
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey
MR Author ID: 1093770
ORCID: 0000-0001-6356-5699
Email: buckmaster@math.princeton.edu

Vlad Vicol
Affiliation: Courant Institute for Mathematical Sciences, New York University, New York, New York
MR Author ID: 846012
ORCID: setImmediate$0.00243841196800898$2
Email: vicol@cims.nyu.edu

Received by editor(s): August 25, 2020
Published electronically: November 2, 2020
Additional Notes: The first author was supported by the NSF grant DMS-1900149 and a Simons Foundation Mathematical and Physical Sciences Collaborative Grant.
The second author was supported by the NSF grant CAREER DMS–1911413.
Article copyright: © Copyright 2020 American Mathematical Society