On Admissibility Criteria for Weak Solutions of the Euler Equations

Lellis, Camillo De; Székelyhidi, László

doi:10.1007/s00205-008-0201-x

Cited by 480 publications

(795 citation statements)

References 21 publications

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“…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: 91%

“…Observe that for the particular choice ̺ ≡ ̺, the problem (1.1-1.3) reduces to the "damped" Euler system with zero pressure. In view of the recent ground-breaking results by DeLellis and Székelyhidi [12], [10], [11] based on the method of convex integration, such a system is ill-posed in the class of weak solutions, meaning it admits infinitely many solutions for any initial data.…”

Section: Weak Solutionsmentioning

confidence: 99%

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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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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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Section: Weak Solutionsmentioning

confidence: 91%

Section: Weak Solutionsmentioning

confidence: 99%

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%

“…-there are no isentropic gases, and non-uniqueness is due to an error in this basic (but unrealistic) model in continuum mechanics and -the admissibility criteria used in [1] are inadequate and do not reflect physical reality.…”

Section: Introductionmentioning

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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“…However, the construction of [DLSz10] does not exclude the existence of rough initial data for which the Cauchy problem associated with Euler equations (5.1) have an infinite set of dissipative solutions. In fact, it is observed in [DLSz10] that any weak solution with a non-increasing energy,…”

Section: Thementioning

confidence: 99%

Stability and spectral convergence of Fourier method for nonlinear problems: on the shortcomings of the $$2/3$$ 2 / 3 de-aliasing method

Bardos

Tadmor

2014

Numer. Math.

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Abstract. The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers' equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two main flavors. One is the spectral Fourier method. The other is the 2/3 pseudo-spectral Fourier method, where one removes the highest 1/3 portion of the spectrum; this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors. Two main themes are discussed in this paper. First, we prove that as long as the underlying exact solution has a minimal C 1+α spatial regularity, then both the spectral and the 2/3 pseudo-spectral Fourier methods are stable. Consequently, we prove their spectral convergence for smooth solutions of the inviscid Burgers equation and the incompressible Euler equations. On the other hand, we prove that after a critical time at which the underlying solution lacks sufficient smoothness, then both the spectral and the 2/3 pseudo-spectral Fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations. In particular, after shock formation in inviscid Burgers' equation, the total variation of bounded (pseudo-) spectral Fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3D incompressible Euler equations: the limiting Fourier solution is shown to enforce L 2 -energy conservation, and the contrast with energy dissipating Onsager solutions is reflected through spurious oscillations.

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On Admissibility Criteria for Weak Solutions of the Euler Equations

Cited by 480 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

Stability and spectral convergence of Fourier method for nonlinear problems: on the shortcomings of the $$2/3$$ 2 / 3 de-aliasing method

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