A counterexample to well-posedness of entropy solutions to the compressible Euler system

We consider a general Euler-Korteweg-Poisson system in R 3 , supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any sufficiently smooth initial data including the case of a vanishing initial density -the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solutions satisfying the energy inequality. Finally, we establish the weak-strong uniqueness property in a class of solutions without vacuum.

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“…Similarly, r need not be bounded below away from zero. Thus Lemma 3.1 can be interpreted as a singular version of similar results in [8], [11].…”

Section: Proof Of Theorem 21 By Convex Integrationmentioning

confidence: 75%

“…Chiodaroli [8] obtained similar illposedness results for the compressible Euler system using a "nonconstant" coefficient version of the method of [11]; later the method was further extended in [9] in order to attack the more complex Euler-Fourier system. The main idea, elaborated in [9], is to consider the Helmholtz decomposition…”

Section: Weak Solutionsmentioning

confidence: 86%

Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

Donatelli

Feireisl

Marcati

2014

Communications in Partial Differential Equations

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“…Unfortunately, this generality is also its main disadvantage. Specifically, the beautiful result of de Lellis & Szekelyhidi [1] (see in addition the recent article of Chiodaroli [9]) tells us: Theorem 4.1 seems to reject (at least in its present form) the 'thermodynamic admissibility criterion' given by inequality (3.1). What can be said for the other two?…”

Section: A Comparison Of Admissibility Criteriamentioning

confidence: 98%

“…Capillarity will penalize initial oscillations in density, viscosity will penalize oscillations in the multi-dimensional contact discontinuities which are the heart of the non-uniqueness example of Chiodaroli [9]. But there is a caveat: viscosity must be strong enough to dominate capillarity.…”

Section: Implications For Non-uniqueness Of Euler Equationsmentioning

confidence: 99%

Admissibility of weak solutions for the compressible Euler equations, n ≥ 2

Slemrod

2013

Phil. Trans. R. Soc. A.

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This paper compares three popular notions of admissibility for weak solutions of the compressible isentropic Euler equations of gas dynamics: (i) the viscosity criterion, (ii) the entropy inequality (the thermodynamically admissible isentropic solutions), and (iii) the viscosity–capillarity criterion. An exact summation of the Chapman–Enskog expansion for Grad’s moment system suggests that it is the third criterion that is representing the kinetic theory of gases. This, in turn, may shed some light on the ability to recover weak solutions of the Euler equations via a hydrodynamic limit.

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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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A counterexample to well-posedness of entropy solutions to the compressible Euler system

Cited by 127 publications

References 21 publications

Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

Admissibility of weak solutions for the compressible Euler equations, n ≥ 2

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

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