A Proof of Onsager's Conjecture

By Philip Isett. 2016

499 citations recorded by the source corpus.

For any $\alpha 1/3$ due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent $\alpha = 1/3$ marks the threshold for conservation of energy for weak solutions in the class $L_t^\infty C_x^\alpha$. The previous best results were solutions in the class $C_tC_x^\alpha$ for $\alpha < 1/5$, due to the author, and in the class $L_t^1 C_x^\alpha$ for $\alpha < 1/3$ due to Buckmaster, De Lellis and Szekelyhidi, both based on the method of convex integration developed for the incompressible Euler equations by De Lellis and Szekelyhidi. The present proof combines the method of convex integration and a new "gluing approximation" technique. The convex integration part of the proof relies on the "Mikado flows" introduced by [Daneri, Szekelyhidi] and the framework of estimates developed in the author's previous work.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/8d9441664dba45df51ec011665c20f4b53f1c8a1

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In Applied Mathematics

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Citations in the corpus

Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1 (2018, methodology): “[2] T. Buckmaster, C. De Lellis, P. Isett, and L. Székelyhidi, Jr. Anomalous dissipation for 1/5-Hölder Euler flows.”
On the energy equality for the 3D Navier–Stokes equations (2018, methodology): “In this direction, energy conservation for C0,1/3solutions of the Euler equations has been recently addressed in the work of Isett [32] and also Buckmaster, De Lellis, Székelyhidi, and Vicol [11].”
Onsager's Conjecture for Admissible Weak Solutions (2017, methodology): “nent 1=10. After a series of important partial results improving the threshold and the techniques from several points of view, cf. [Ise13a, BDLS13, BDLISJ15, Buc15, BDLS16], in his recent paper Isett [Ise16] has been able to nally reach the Onsager exponent 1=3. The proof of Date: January 31, 2017. ”
Nonuniqueness of weak solutions to the Navier-Stokes equation (2017, unknown): “1)) via convex integration methods [47, 52, 2, 17, 1, 3], leading to the resolution of the conjecture by Isett [25, 26], using a key ingredient by Daneri and Székelyhidi Jr.”
Convex integration and phenomenologies in turbulence (2019, methodology): “We discuss the papers on which Isett’s construction relies (e.g. the Mikado flows by Daneri-Székelyhidi Jr. [44]) and the main ideas in Isett’s work [102].”
Non-uniqueness of weak solutions to hyperviscous Navier–Stokes equations: on sharpness of J.-L. Lions exponent (2018, methodology): “The schemes in [5, 24] are based on the convex integration framework in Hölder spaces for the Euler equations, introduced by De Lellis and Székelyhidi in [12], subsequently refined in [2,3,10,15], and culminated in the proof of the second half of the Onsager conjecture by Isett in [16]; also see [4]”
Non-uniqueness of Leray solutions of the forced Navier-Stokes equations (2021, methodology): “Their proof introduces the important tool of intermittency and builds on the fundamental works for the Euler equations and the proof of the Onsager conjecture [31, 9] (see [11, 10] for further connections to fluid turbulence).”
Lecture notes on Generalised Hydrodynamics (2019, unknown): “See for instance [130] and [131]6.”
Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks (2019, background): “In fact, in many applied areas the concept of turbulence becomes important, which from a mathematical point of view deals with PDEs in regimes where neither regularity nor stability are observed, a fact with broad conceptual ramifications [32].”
Nonuniqueness of Weak Solutions to the SQG Equation (2016, methodology): “We remark that recently, [Ise16] proved the full Onsager’s conjecture for the 3-D Euler equations, employing a novel technique involving gluing exact solutions to the Euler equations, along with the use of Mikado flows, introduced in [DSJ16] as a replacement to Beltrami waves.”
Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier–Stokes equations: Existence and nonuniqueness (2021, methodology): “This method has already led to a number of groundbreaking results concerning the incompressible Euler equations, culminating in the proof of Onsager’s conjecture by Isett [Ise18] and by Buckmaster, De Lellis, Székelyhidi Jr. and Vicol [BDLSV19].”