Convex integration and phenomenologies in turbulence

Tristan Buckmaster, Vlad Vicol

Research output: Contribution to journalArticle

Abstract

In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier–Stokes equations. These results build on the groundbreaking works of De Lellis and Székelyhidi Jr., who extended Nash’s fundamental ideas on C 1 flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration, have fundamental analogies with the phenomenological theories of hydrodynamic turbulence [51, 54, 55, 200]. Mathematical problems arising in turbulence (such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions. First, we give an elementary construction of nonconservative Cx;t 0C weak solutions of the Euler equations, first proven by De Lellis–Székelyhidi Jr. [52, 53]. Second, we present Isett’s [108] recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work [21] of De Lellis–Székelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class Cx;t 1=3- are constructed, attaining any energy profile. Third, we give a concise proof of the authors’ recent result [23], which proves the existence of infinitely many weak solutions of the Navier–Stokes in the regularity class Ct 0L2+ x \ Ct 0Wx 1;1C. We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.

Original languageEnglish (US)
Pages (from-to)143-263
Number of pages121
JournalEMS Surveys in Mathematical Sciences
Volume6
Issue number1-2
DOIs
StatePublished - 2019

Keywords

  • Convex integration
  • Hydrodynamic turbulence
  • Incompressible Euler
  • Incompressible Navier–Stokes
  • Intermittency
  • Inviscid limit
  • Non-uniqueness
  • Onsager’s conjecture
  • Weak solutions

ASJC Scopus subject areas

  • Mathematics(all)

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