Regularity and Energy Conservation for the Compressible Euler Equations

Feireisl, Eduard; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka; Wiedemann, Emil

doi:10.1007/s00205-016-1060-5

Cited by 90 publications

(128 citation statements)

References 22 publications

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“…≤ C feels rather artificial and is not in the p ∈ C 2 result from [16]. We will now focus on finding conditions on ρ for different L q norms that will control this term.…”

Section: Energy Conservation Assuming Hölder Continuity Of the Pressurementioning

confidence: 99%

Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum

et al. 2020

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We consider the compressible isentropic Euler equations on T d × [0, T ] with a pressure law p ∈ C 1,γ−1 , where 1 ≤ γ < 2. This includes all physically relevant cases, e.g. the monoatomic gas. We investigate under what conditions on its regularity a weak solution conserves the energy. Previous results have crucially assumed that p ∈ C 2 in the range of the density, however, for realistic pressure laws this means that we must exclude the vacuum case. Here we improve these results by giving a number of sufficient conditions for the conservation of energy, even for solutions that may exhibit vacuum: Firstly, by assuming the velocity to be a divergence-measure field; secondly, imposing extra integrability on 1/ρ near a vacuum; thirdly, assuming ρ to be quasi-nearly subharmonic near a vacuum; and finally, by assuming that u and ρ are Hölder continuous. We then extend these results to show global energy conservation for the domain Ω × [0, T ] where Ω is bounded with a C 2 boundary. We show that we can extend these results to the compressible Navier-Stokes equations, even with degenerate viscosity. MSC (2010): 35Q31 (PRIMARY); 35Q10, 35L65, 76N10.

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“…≤ C feels rather artificial and is not in the p ∈ C 2 result from [16]. We will now focus on finding conditions on ρ for different L q norms that will control this term.…”

Section: Energy Conservation Assuming Hölder Continuity Of the Pressurementioning

confidence: 99%

Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum

et al. 2020

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“…Chen and Glimm [6]. Here, we are motivated by the proof of the one sided implication in Onsager's conjecture by Constantin, E, and Titi [11], and its subsequent generalization to the compressible Euler system in [17]. In particular, we use the fact the Besov functions can be regularized by convolution kernels and the resulting commutators with non-linear superpositions can be effectively controlled.…”

Section: Introductionmentioning

confidence: 99%

On uniqueness of dissipative solutions to the isentropic Euler system

Feireisl

Ghoshal

Jana

2019

Communications in Partial Differential Equations

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The dissipative solutions can be seen as a convenient generalization of the concept of weak solution to the isentropic Euler system. They can be seen as expectations of the Young measures associated to a suitable measure-valued solution of the problem. We show that dissipative solutions coincide with weak solutions starting from the same initial data on condition that: (i) the weak solution enjoys certain Besov regularity; (ii) the symmetric velocity gradient of the weak solution satisfies a one-sided Lipschitz bound.

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“…[3].It is most natural to ask whether the analogue of the Onsager conjecture is valid for other systems of conservation laws. Indeed, there has been some intensive recent work extending the Onsager conjecture for other physical systems, in the absence of physical boundaries, see, e.g., [1,12,15,16,20] and references therein. In this paper we consider systems of conservation laws with physical boundaries.…”

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confidence: 98%

On the Extension of Onsager’s Conjecture for General Conservation Laws

et al. 2018

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The aim of this work is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this work is the "universality" of the Onsager exponent, α > 1/3, concerning the regularity of the solutions, say in C 0,α , that guarantees the conservation of the generalized entropy; regardless of the structure of the genuine nonlinearity in the underlying system.In this work we aim at extending and proving the Onsager conjecture for a class of conservation laws that admit a generalized entropy. Roughly speaking, the Onsager conjecture [18] states that weak solutions of the three-dimensional Euler equations of inviscid incompressible flows conserve energy if the velocity field u ∈ C 0,α , for α > 1 3 , and that the critical exponent α = 1 3 is sharp. This conjecture has been the subject of intensive investigation for the last two decades. The sufficient condition direction was proved by Eyink [14] for the case when α > 1 2 . Later, a complete proof was established by Constantin, E and Titi [9] (see also [8]) under slightly weaker regularity assumptions on the solution which involve a similar exponent α > 1 3 . Duchon and Robert [13] have shown, under similar sufficient conditions to those in [9], a local version of the conservation of energy. It is worth mentioning that the above results are established in the absence of physical boundaries, i.e., periodic boundary conditions or the whole space. However, due to the well recognized dominant role of the boundary in the generation of turbulence (cf.[4] and references therein) it seems very reasonable to investigate the analogue of the Onsager conjecture in bounded domains. Indeed, for the three-dimensional Euler equations in a smooth bounded domain Ω, subject to no-normal flow (slip) boundary conditions, it has been shown in [5] that a weak solution conserves the energy provided the velocity field u ∈ C 0,α (Ω) , for α > 1 3 , (see also [19] for the case of the upper-half space under stronger conditions on the pressure term). A local version, analogue to that of [13], was established recently in [6] under slightly weaker conditions to those in [5], but at the expense of additional sufficient conditions concerning the vanishing behavior of the energy flux near the boundary.Showing the sharpness of the exponent α = 1 3 in Onsager's conjecture turns out to be much more subtle. This direction has been underlined by a series of contributions (cf. Isett [17], Buckmaster, De Lellis , Székelyhidi and Vicol [7] and references therein) where weak solutions, u ∈ C 0,α , with α < 1 3 , that dissipate energy were constructed using the convex integration machinery. Notice, however, that there exists a family of weak solutions to the three-dimensional Euler equations, that are not more regular than L 2 , and which conserve the energy, cf. [3].It is most natural to ask whether the analogue of the Onsager conjecture is valid for other systems of conservation laws. Indeed, there has been some intensive recent work extendi...

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Regularity and Energy Conservation for the Compressible Euler Equations

Cited by 90 publications

References 22 publications

Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum

Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum

On uniqueness of dissipative solutions to the isentropic Euler system

On the Extension of Onsager’s Conjecture for General Conservation Laws

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