Energy Conservation in Two-dimensional Incompressible Ideal Fluids

Cheskidov, Alexey; Filho, Milton C. Lopes; Lopes, Helena J. Nussenzveig; Shvydkoy, Roman

doi:10.1007/s00220-016-2730-8

Cited by 43 publications

(102 citation statements)

References 19 publications

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“…The case d = 2 is different, because of the absence of vortex-stretching. This implies strong bounds on enstrophy for Leray solutions in d = 2, even with initial vorticity ω 0 ∈ L p only for p < 2, and an essential improvement of the energy dissipation bounds in our Lemma 1 for d = 2 [13].…”

Section: Introductionmentioning

confidence: 78%

An Onsager singularity theorem for Leray solutions of incompressible Navier–Stokes

Drivas

Eyink

2019

Nonlinearity

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We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus T d , assuming that the solutions have norms for Besov space B σ,∞ 3 (T d ), σ ∈ (0, 1], that are bounded in the L 3 -sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form O(ν (3σ−1)/(σ+1) ), vanishing as ν → 0 if σ > 1/3. A consequence is that Onsagertype "quasi-singularities" are required in the Leray solutions, even if the total energy dissipation vanishes in the limit ν → 0, as long as it does so sufficiently slowly. We also give two sufficient conditions which guarantee the existence of limiting weak Euler solutions u which satisfy a local energy balance with possible anomalous dissipation due to inertial-range energy cascade in the Leray solutions. For σ ∈ (1/3, 1) the anomalous dissipation vanishes and the weak Euler solutions may be spatially "rough" but conserve energy.

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Section: Introductionmentioning

confidence: 78%

An Onsager singularity theorem for Leray solutions of incompressible Navier–Stokes

Drivas

Eyink

2019

Nonlinearity

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“…Finally, we comment on the extension from the flat torus to the whole space, which is crucial to address the fundamental question of the conservation of the energy: it allows us to extend from the two-dimensional torus to the whole space the result of [12] on the conservation of kinetic energy for solutions of the Euler equations obtained as limit of vanishing viscosity when the initial vorticity is in L p . Indeed, as already noticed in [14], the main issue in extending the result of [12] to the whole space is to obtain global strong convergence in C(L 2 ) of the velocity. Due to the lack of compact embedding this cannot be obtained by using the Aubin-Lions lemma, but it is obtained by exploiting a Serfati-type formula [37], which in turn requires the strong convergence of the vorticities.…”

Section: 7)mentioning

confidence: 99%

“…Indeed, the conservation of the L p -norm of the vorticity is a consequence of (2.5) and the fact that the flow X t,0 (•) is measure-preserving, while the conservation of the energy is one of the main results in [12].…”

Section: Quantitative Strong Convergence Of the Vorticitymentioning

confidence: 99%

Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

Ciampa

Crippa

Spirito

2021

Arch Rational Mech Anal

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In this paper we prove the uniform-in-time $$L^p$$ L p convergence in the inviscid limit of a family $$\omega ^\nu $$ ω ν of solutions of the 2D Navier–Stokes equations towards a renormalized/Lagrangian solution $$\omega $$ ω of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $$\omega ^{\nu }$$ ω ν to $$\omega $$ ω in $$L^p$$ L p . Finally, we show that solutions of the Euler equations with $$L^p$$ L p vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.

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“…Stronger results in two-dimensions are known without boundary. In particular, there is an interesting result in [6] showing that if any weak Euler solution is obtained in the vanishing viscosity limit of strong Navier-Stokes solutions in T 2 , then it conserves the energy provided only that the initial vorticity is ω 0 ∈ L p (T 2 ) with p > 1. Recall that in 2D, in view of the scaling invariant embedding W 1,3/2 ⊂ B 1/3,∞ 3 , the Onsager threshold corresponds to the integrability L p with p > 3/2 for the vorticity.…”

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confidence: 99%