Onsager's conjecture in bounded domains for the conservation of entropy and other companion laws

Bardos, Claude; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka; Titi, Edriss S.; Wiedemann, Emil

doi:10.1098/rspa.2019.0289

Cited by 15 publications

(11 citation statements)

References 29 publications

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“…In the current paper we study relaxation of the C 2 assumption on the nonlinearities in the context of general conservation laws and their companion quantities. We prove an analogue of Theorem 1.1 in [19], using the function space framework of [5]. Our main result is the following.…”

Section: Introductionmentioning

confidence: 92%

“…to prove local energy conservation for the incompressible Euler, see also [15,22]. Inspired by (2.2), Bardos et al [5] introduce the Besov-V MO type space B 1/3 3,V MO . We use here this construction with suitable modifications.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…require Besov regularity only with respect to some variables). Furthermore, in [5] a similar extension of Onsager's conjecture to general conservation laws is done in presence of physical boundaries, introducing a more optimal Besov-V MO type space, see also [4,6,7].…”

Section: Introductionmentioning

confidence: 99%

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On entropy conservation for general systems of conservation laws

Dębiec

2019

Preprint

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Section: Introductionmentioning

confidence: 92%

Section: Preliminaries and Notationmentioning

confidence: 99%

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On entropy conservation for general systems of conservation laws

Dębiec

2019

Preprint

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“…We will get back to the vacuum problem in the next section. More generally, the Taylor expansion strategy applies to essentially any system of conservation laws that possesses an entropy [6,7,33]. To illustrate the point, let…”

Section: General Conservation Lawsmentioning

confidence: 99%

Conserved quantities and regularity in fluid dynamics

Wiedemann¹

2020

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This is a set of lecture notes for the 2019 EMS School in Applied Mathematics held in Kácov, Czech Republic.Conserved or dissipated quantities, like energy or entropy, are at the heart of the study of many classes of time-dependent PDEs in connection with fluid mechanics. This is the case, for instance, for the Euler and Navier-Stokes equations, for systems of conservation laws, and for transport equations. In all these cases, a formally conserved quantity may no longer be constant in time for a weak solution at low regularity. The delicate interplay between regularity and conservation of the respective quantity relates to renormalisation in the DiPerna-Lions theory of transport and continuity equations, and to Onsager's conjecture in the realm of ideal incompressible fluids. We will review the classical commutator methods of DiPerna-Lions and Constantin-E-Titi, and then proceed to more recent results. CONTENTS 17 7. Degenerate cases 19 References 20

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“…Very recently, the hypothesis π ∈ C 2 was improved to π ∈ C 1,γ−1 with 1 ≤ γ < 2 by Akramov-Debiec-Skipper-Wiedemann in [1]. Moreover, under the density ̺ ∈ L ∞ ([0, T ] × T d ), sufficient conditions for weak solutions of system (1.2) and (1.3) to conserve the energy can be found in [2,21,24,29].…”

Section: Introductionmentioning

confidence: 99%

The role of density in the energy conservation for the isentropic compressible Euler equations

Wang¹,

Ye²,

Yu³

2021

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In this paper, we study Onsager's conjecture on the energy conservation for the isentropic compressible Euler equations via establishing the energy conservation criterion involving the density ̺ ∈ L k (0, T ; L l (T d )). The motivation is to analysis the role of the integrability of density of the weak solutions keeping energy in this system, since almost all known corresponding results require ̺ ∈ L ∞ (0, T ; L ∞ (T d )). Our results imply that the lower integrability of the density ̺ means that more integrability of the velocity v are necessary in energy conservation and the inverse is also true. The proof relies on the Constantin-E-Titi type and Lions type commutators on mollifying kernel.

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Onsager's conjecture in bounded domains for the conservation of entropy and other companion laws

Cited by 15 publications

References 29 publications

On entropy conservation for general systems of conservation laws

On entropy conservation for general systems of conservation laws

Conserved quantities and regularity in fluid dynamics

The role of density in the energy conservation for the isentropic compressible Euler equations

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