TY - JOUR

T1 - Convex integration and phenomenologies in turbulence

AU - Buckmaster, Tristan

AU - Vicol, Vlad

N1 - Funding Information:
Acknowledgements. The work of T. B. has been partially supported by the NSF grant DMS-1600868. V. V. was partially supported by the NSF grant CAREER DMS-1911413. The authors are grateful to Raj Beekie, Theodore Drivas, Matthew Novack, and Lenya Ryzhik for suggestions and stimulating discussions concerning aspects of this review. The authors would also like to thank the anonymous referees for detailed comments and suggestions for improvement of this manuscript.
Publisher Copyright:
© European Mathematical Society

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Convex integration

KW - Hydrodynamic turbulence

KW - Incompressible Euler

KW - Incompressible Navier–Stokes

KW - Intermittency

KW - Inviscid limit

KW - Non-uniqueness

KW - Onsager’s conjecture

KW - Weak solutions

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U2 - 10.4171/EMSS/34

DO - 10.4171/EMSS/34

M3 - Article

AN - SCOPUS:85082736643

VL - 6

SP - 143

EP - 263

JO - EMS Surveys in Mathematical Sciences

JF - EMS Surveys in Mathematical Sciences

SN - 2308-2151

IS - 1-2

ER -