A heat flow approach to Onsager’s conjecture for the Euler equations on manifolds

Isett, Philip; Oh, Sung-Jin

doi:10.1090/tran/6549

Cited by 17 publications

(19 citation statements)

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“…Coarsegraining operations in GR have attracted recent interest also because of problems in cosmology and in the interpretation of cosmological observations, and much of this parallel work [104,105] should carry over to generalrelativistic turbulence. Here we may note that an Onsager singularity theorem has already been proved for incompressible fluid turbulence on general compact Riemannian manifolds, by exploiting a coarse-graining regularization defined with a heat kernel smoothing [106].…”

Section: Summary and Future Directionsmentioning

confidence: 93%

Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence

Eyink¹,

Drivas²

2018

Phys. Rev. X

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We develop first-principles theory of relativistic fluid turbulence at high Reynolds and Péclet numbers. We follow an exact approach pioneered by Onsager, which we explain as a non-perturbative application of the principle of renormalization-group invariance. We obtain results very similar to those for non-relativistic turbulence, with hydrodynamic fields in the inertial-range described as distributional or "coarse-grained" solutions of the relativistic Euler equations. These solutions do not, however, satisfy the naive conservation-laws of smooth Euler solutions but are afflicted with dissipative anomalies in the balance equations of internal energy and entropy. The anomalies are shown to be possible by exactly two mechanisms, local cascade and pressure-work defect. We derive "4/5th-law"-type expressions for the anomalies, which allow us to characterize the singularities (structure-function scaling exponents) required for their non-vanishing. We also investigate the Lorentz covariance of the inertial-range fluxes, which we find is broken by our coarse-graining regularization but which is restored in the limit that the regularization is removed, similar to relativistic lattice quantum field theory. In the formal limit as speed of light goes to infinity, we recover the results of previous non-relativistic theory. In particular, anomalous heat input to relativistic internal energy coincides in that limit with anomalous dissipation of non-relativistic kinetic energy.

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Section: Summary and Future Directionsmentioning

confidence: 93%

Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence

Eyink¹,

Drivas²

2018

Phys. Rev. X

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“…Other results that help draw attention to this point of view are the works of [11,16,21]. This local perspective on the problem is emphasized by the local character of Theorems 1.1 and 1.2, and by the improvements in our construction that allow us to achieve this localization.…”

Section: E(t+ T)−e(t)|mentioning

confidence: 93%

On Nonperiodic Euler Flows with Hölder Regularity

Isett

2016

Arch Rational Mech Anal

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In (Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013), the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space L ∞ t B 1/3 3,∞ due to low regularity of the energy profile. This paper is the first and main paper in a series of two, the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of the present paper shows that any given smooth Euler flow can be perturbed in C 1/5−ε t,x on any pre-compact subset of R × R 3 to violate energy conservation. Furthermore, the perturbed solution is no smoother than C. As a corollary of this theorem, we show the existence of nonzero C 1/5−ε t,x solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting.

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“…This result allows for the possibility that the failure of energy conservation in Conjecture 2 may also hold in the endpoint case α = 1/3, and [CCFS08] provides an example that suggests that fluctuations in kinetic energy should indeed be possible for α = 1/3. We refer also to [DR00,IO16] for extensions of these results and alternative proofs.…”

Section: Part I Introductionmentioning

confidence: 99%

A proof of Onsager's conjecture

2018

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For any α < 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class CtC αx that have nonempty, compact support in time on R × T 3 and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for α > 1/3 due to [Eyi94, CET94], solves Onsager's conjecture that the exponent α = 1/3 marks the threshold for conservation of energy for weak solutions in the class L ∞ t C α x . The previous best results were solutions in the class CtC αx for α < 1/5, due to the author, and in the class L 1 t C α x for α < 1/3 due to Buckmaster, De Lellis and Székelyhidi, both based on the method of convex integration developed for the incompressible Euler equations in [DLS13b, DLS14]. 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 in [DS16] and the framework of estimates developed in the author's previous work. Contents I Introduction

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A heat flow approach to Onsager’s conjecture for the Euler equations on manifolds

Cited by 17 publications

References 20 publications

Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence

Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence

On Nonperiodic Euler Flows with Hölder Regularity

A proof of Onsager's conjecture

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