Weak-strong uniqueness for measure-valued solutions of some compressible fluid models

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

doi:10.1088/0951-7715/28/11/3873

Cited by 97 publications

(105 citation statements)

References 46 publications

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“…See [14]. We introduce the concept of dissipative measure-valued solution to the system (1.1) -(1.2) in the spirit of [8] and [9]. Denition 6.…”

Section: Remark 4 In the Denition 3 The Young Measures Are Dened Formentioning

confidence: 99%

“…In the framework of dissipative measure-valued solution (in the spirit of [9]- [8]) we dene the functional …”

Section: Topical Problems Of Fluid Mechanics 3 ______________________mentioning

confidence: 99%

“…compressible Euler and Navier-Stokes equations [13]. The measure-valued solution to the non-Newtonian case was proved by Novotný, Ne£asová [14].Recently, weak-strong uniqueness for measure-valued solutions of isentropic Euler equations were proved in [8]. Inspired by previous results, the concept of dissipative measure-valued solution was nally applied to the barotropic compressible Navier-Stokes system [9].…”

mentioning

confidence: 99%

“…Recently, weak-strong uniqueness for measure-valued solutions of isentropic Euler equations were proved in [8]. Inspired by previous results, the concept of dissipative measure-valued solution was nally applied to the barotropic compressible Navier-Stokes system [9].…”

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confidence: 99%

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Note on the Problem of Dissipative Measure-Valued Solutions to the Compressible Non-Newtonian System

Baba¹,

Caggio²,

Ducomet³

et al. 2017

Topical Problems of Fluid Mechanics 2017

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We introduce a dissipative measure-valued solution to the compressible non-Newtonian system. We generalized a result given by Novotný, Ne£asová [14]. We derive a relative entropy inequality for measure-valued solution as an extension of the "classical" entropy inequality introduced by Dafermos 1Formulation of the problemWe consider measure-valued solutions of the compressible non-Newtonian system. The advantage of measure-valued solutions is the property that in many cases, the solutions can be obtained from weakly convergent sequences of approximate solutions. Measure-valued solutions for systems of hyperbolic conservations laws were initially introduced by DiPerna [3]. He used Young measures to pass to limit in the articial viscosity term. In the case of the incompressible Euler equations, DiPerna and Majda [4] also proved global existence of measure-valued solutions for any initial data with nite energy. They introduced generalized Young measures to take into account oscillation and concentration phenomena. Thereafter the existence of measure-valued solutions was nally shown for further models of uids, e.g. compressible Euler and Navier-Stokes equations [13]. The measure-valued solution to the non-Newtonian case was proved by Novotný, Ne£asová [14].Recently, weak-strong uniqueness for measure-valued solutions of isentropic Euler equations were proved in [8]. Inspired by previous results, the concept of dissipative measure-valued solution was nally applied to the barotropic compressible Navier-Stokes system [9]. It is a generalization of classical measure-valued solution.The motion of the uid is governed by the following system of equationswhere is the mass density and u is the velocity eld, functions of the spatial position x ∈ R 3 and the time t ∈ R. The scalar function p is termed pressure, given function of the density. In particular, we consider the isothermal case, namely p = λ , with λ > 0 a constant. The stress tensor is given by S ij = βu l,l δ ij

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“…See [14]. We introduce the concept of dissipative measure-valued solution to the system (1.1) -(1.2) in the spirit of [8] and [9]. Denition 6.…”

Section: Remark 4 In the Denition 3 The Young Measures Are Dened Formentioning

confidence: 99%

“…In the framework of dissipative measure-valued solution (in the spirit of [9]- [8]) we dene the functional …”

Section: Topical Problems Of Fluid Mechanics 3 ______________________mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Note on the Problem of Dissipative Measure-Valued Solutions to the Compressible Non-Newtonian System

Baba¹,

Caggio²,

Ducomet³

et al. 2017

Topical Problems of Fluid Mechanics 2017

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“…On the other hand, measure-valued solutions have been criticized for being too weak, as is apparent from their obvious non-uniqueness (but see the weak-strong uniqueness results in [2,14]). However, recent results by the first three authors jointly with De Lellis [3,4,5] have demonstrated that for the compressible Euler equations (1.1) even entropy solutions (weak solutions satisfying a suitable entropy inequality) may not be unique, thus raising anew the problem of a correct notion of solutions for (1.1).…”

Section: Introductionmentioning

confidence: 99%

$${\mathcal {A}}$$ A -free rigidity and applications to the compressible Euler system

Chiodaroli

Feireisl

Kreml

et al. 2017

Annali di Matematica

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Can every measure-valued solution to the compressible Euler equations be approximated by a sequence of weak solutions? We prove that the answer is negative: Generalizing a well-known rigidity result of Ball and James to a more general situation, we construct an explicit measure-valued solution for the compressible Euler equations which can not be generated by a sequence of distributional solutions. We also give an abstract necessary condition for measure-valued solutions to be generated by weak solutions, relying on work of Fonseca and Müller.While a priori it is not unexpected that not every measure-valued solution arises from a sequence of weak solutions, it is noteworthy that this observation in the compressible case is in contrast with the incompressible situation, where every measurevalued solution can be approximated by weak solutions, as shown by Székelyhidi and Wiedemann.

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On the low Mach number limit of the compressible viscous micropolar fluid model

Yang

et al. 2022

Math Methods in App Sciences

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In this paper, we consider the low Mach number limit problem for the compressible viscous micropolar fluid model based on the concept of dissipative measure‐valued solutions. We prove that the dissipative measure‐valued solutions of the compressible micropolar fluid model converge to the smooth solution of the incompressible micropolar system in the case of well‐prepared initial data when the Mach number tends to zero.

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Weak-strong uniqueness for measure-valued solutions of some compressible fluid models

Cited by 97 publications

References 46 publications

Note on the Problem of Dissipative Measure-Valued Solutions to the Compressible Non-Newtonian System

Note on the Problem of Dissipative Measure-Valued Solutions to the Compressible Non-Newtonian System

$${\mathcal {A}}$$ A -free rigidity and applications to the compressible Euler system

On the low Mach number limit of the compressible viscous micropolar fluid model

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