On the energy equality for the 3D Navier–Stokes equations

Berselli, Luigi C.; Chiodaroli, Elisabetta

doi:10.1016/j.na.2019.111704

Cited by 52 publications

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References 39 publications

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“…In references [1] and [2] the authors proved the energy equality under conditions of type u ∈ L p (0, T ; W 1, q (Ω)) , (2.1) instead of type (1.4). However the above Shinbrot's number θ(p, r) was extended to this new situation by appealing to the sharp Sobolev's embedding theorem Hence we will restrict the conditions of type (2.1) to the above q−range.…”

Section: Comparison With Previous Resultsmentioning

confidence: 99%

On the Shinbrot’s criteria for energy equality to Newtonian fluids: A simplified proof, and an extension of the range of application

Veiga

Yang

2020

Nonlinear Analysis

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We show that the classical Shinbrot's criteria to guarantee that a Leray-Hopf solution satisfies the energy equality follows trivially from the L 4 ((0 , T ) × Ω)) Lions-Prodi particular case. Moreover we extend Shinbrot's result to space coefficients r ∈ (3, 4) . In this last case our condition coincides with Shinbrot condition for r = 4, but for r < 4 it is more restrictive than the classical one, 2/p + 2/r = 1 . It looks significant that in correspondence to the extreme values r = 3 and r = ∞, and just for these two values, the conditions become respectively u ∈ L ∞ (L 3 ) and u ∈ L 2 (L ∞ ), which imply regularity by appealing to classical Ladyzhenskaya-Prodi-Serrin (L-P-S) type conditions. However, for values r ∈ (3, ∞) the L-P-S condition does not apply, even for the more demanding case 3 < r < 4 . The proofs are quite trivial, by appealing to interpolation, with L ∞ (L 2 ) in the first case and with L 2 (L 6 ) in the second case. The central position of this old classical problem in Fluid-Mechanics, together with the simplicity of the proofs (in particular the novelty of the second result) looks at least curious. This may be considered a merit of this very short note.Mathematics Subject Classification: 35Q30, 76A05, 76D03.

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Section: Comparison With Previous Resultsmentioning

confidence: 99%

On the Shinbrot’s criteria for energy equality to Newtonian fluids: A simplified proof, and an extension of the range of application

Veiga

Yang

2020

Nonlinear Analysis

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“…This happens, in particular, to the quite strong results obtained by L.C. Berselli and E. Chiodaroli in reference [1]. One of our aim is to solve, positively, this anomaly.…”

Section: )mentioning

confidence: 78%

“…After the above 1974 Shinbrot's paper [12] new results, enjoying a better SH measure, appeared, see [1] for references. Application of the SH condition to these stronger results have brought to light a negative feature of the SH condition.…”

Section: )mentioning

confidence: 99%

“…This is the aim of the next two sections. In particular, in section 3, we recall Berselli and Chiodaroli's Theorem 5.1, in reference [1]. This reference gave us motivation, and lines of reasoning, leading to the present extension.…”

mentioning

confidence: 93%

“…It seems suitable to inform the interested readers that, as far as we know from the authors, reference [1] is not the final version.…”

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confidence: 99%

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On the energy equality for solutions to Newtonian and non-Newtonian fluids

Veiga

Yang

2019

Nonlinear Analysis

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We are concerned with the energy equality for weak solutions to Newtonian and non-Newtonian incompressible fluids. In particular, the results obtained for non-Newtonian fluids, after restriction to the Newtonian case, equal or improve the known results. Furthermore, a new result below allows an interpretation of the classical Shinbrot condition, coherent to the more recent results in the literature.Mathematics Subject Classification: 35Q30, 76A05, 76D03.

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A general sufficient criterion for energy conservation in the Navier–Stokes system

Wang

2023

Math Methods in App Sciences

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In this paper, we derive an energy conservation criterion based on a combination of velocity and its gradient for the weak solutions of both the homogeneous incompressible Navier-Stokes equations and the general compressible Navier-Stokes equations. For the incompressible case, this class implies almost known corresponding results on periodic domain via either the velocity or its gradient including the famous Lions' energy conservation criterion obtained in [1, Rend. Semin. Mat. Univ. Padova, 30 (1960)]. For the compressible case, this helps us to extend the previously known criteria for the energy conservation of weak solutions from the incompressible fluid to compressible flow and improve the recent results due to Nguyen-Nguyen-Tang in [26, Nonlinearity, 32 (2019)] and Liang in [27, Proc. Roy. Soc. Edinburgh Sect. A, (2020)].

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On the energy equality for the 3D Navier–Stokes equations

Cited by 52 publications

References 39 publications

On the Shinbrot’s criteria for energy equality to Newtonian fluids: A simplified proof, and an extension of the range of application

On the Shinbrot’s criteria for energy equality to Newtonian fluids: A simplified proof, and an extension of the range of application

On the energy equality for solutions to Newtonian and non-Newtonian fluids

A general sufficient criterion for energy conservation in the Navier–Stokes system

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