Absence of Anomalous Dissipation of Energy in Forced Two Dimensional Fluid Equations

Constantin, Peter; Tarfulea, Andrei; Vicol, Vlad

doi:10.1007/s00205-013-0708-7

Cited by 30 publications

(35 citation statements)

References 77 publications

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“…for all 0 ≤ s ≤ t. This is connected to the absence of anomalous dissipation in the critical case, as proven in [12]. A straightforward yet very important consequence is that vanishing viscosity solutions are strongly continuous.…”

Section: Vanishing Viscosity Solutionsmentioning

confidence: 76%

Smooth attractors for weak solutions of the SQG equation with critical dissipation

Zelati¹,

Kalita²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Hölder norms, which complement the existing estimates based on commutator analysis.2000 Mathematics Subject Classification. 35Q35, 35B41, 35B45.

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Section: Vanishing Viscosity Solutionsmentioning

confidence: 76%

Smooth attractors for weak solutions of the SQG equation with critical dissipation

Zelati¹,

Kalita²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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“…Other works have been seeking a priori estimates in the high Reynolds number limit to provide some constraints on the possibilities; see e.g. [10,22,23,26] and the references therein. The work on Onsager's conjecture in 3d can be seen as another kind of consistency check (see e.g.…”

Section: Stationary Solutions and Weak Anomalous Dissipationmentioning

confidence: 99%

Sufficient Conditions for Dual Cascade Flux Laws in the Stochastic 2d Navier–Stokes Equations

Bedrossian

Zelati

Punshon-Smith

et al. 2020

Arch Rational Mech Anal

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We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws capturing the energy flux to large scales and enstrophy flux to small scales for statistically stationary, forced-dissipated 2d Navier-Stokes equations in the large-box limit. These laws should be regarded as 2d turbulence analogues of the 4/5 law in 3d turbulence, predicting a constant flux of energy and enstrophy (respectively) through the two inertial ranges in the dual cascade of 2d turbulence. Conditions implying only one of the two cascades are also obtained, as well as compactness criteria which show that the provided sufficient conditions are not far from being necessary. The specific goal of the work is to provide the weakest characterizations of the "0-th laws" of 2d turbulence in order to make mathematically rigorous predictions consistent with experimental evidence. Contents2010 Mathematics Subject Classification. 35Q30, 60H30, 76F05.

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“…The long time behavior of solutions to the critical SQG equations have been studied in [6,10,14,15,20,34,35]. The first result on the existence of an attractor was obtained recently by Constantin, Tarfulea, and Vicol in [15], where the authors studied the long time dynamics of regular solutions of the forced critical SQG using the nonlinear maximal principle [16].…”

Section: Introductionmentioning

confidence: 99%

Determining modes for the surface quasi-geostrophic equation

Cheskidov

Dai

2018

Physica D: Nonlinear Phenomena

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We introduce a determining wavenumber for the surface quasi-geostrophic (SQG) equation defined for each individual trajectory and then study its dependence on the force. While in the subcritical and critical cases this wavenumber has a uniform upper bound, it may blow up when the equation is supercritical. A bound on the determining wavenumber provides determining modes, which in some sense measure the number of degrees of freedom of the flow, or resolution needed to describe a solution to the SQG equation.

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Absence of Anomalous Dissipation of Energy in Forced Two Dimensional Fluid Equations

Cited by 30 publications

References 77 publications

Smooth attractors for weak solutions of the SQG equation with critical dissipation

Smooth attractors for weak solutions of the SQG equation with critical dissipation

Sufficient Conditions for Dual Cascade Flux Laws in the Stochastic 2d Navier–Stokes Equations

Determining modes for the surface quasi-geostrophic equation

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