Uniqueness and Stability of Riemann Solutions¶with Large Oscillation in Gas Dynamics

Frid, Hermano; Li, Yachun

doi:10.1007/s002200200615

Cited by 65 publications

(54 citation statements)

References 29 publications

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“…measure solutions) in the class of entropy solutions, which may not belong to either BV loc or L ∞ , without specific reference on the method of construction of the solutions. The proof, motivated by [11] and [7]- [9], is heavily based on our explicit approach to calculate normal traces over Lipschitz deformable surfaces, in the generalized GaussGreen theorem, and the product rule for DM-fields. The same arguments clearly also yield the uniqueness and stability of Riemann solutions in the class of entropy solutions for the Euler equations (1.5) and (1.6) for isentropic gas dynamics.…”

Section: Example 12mentioning

confidence: 99%

“…We also refer to Dafermos [10] for the stability of Lipschitz solutions for hyperbolic systems of conservation laws. In [8,9], the uniqueness and stability of Riemann solutions of large oscillation without vacuum (possibly containing shocks) was proved for the 3 × 3 Euler equations, in the class of entropy solutions in L ∞ ∩ BV loc which stay away from vacuum. Another related connection is the recent results on the L 1 -stability of the solutions in L ∞ ∩ BV obtained either by the Glimm scheme [15], the wave front-tracking method, the vanishing viscosity method, or more generally satisfying an additional regularity, with small total variation in x uniformly for all t > 0 (see the recent references cited in Bianchini-Bressan [3] and Dafermos [10]).…”

Section: Example 12mentioning

confidence: 99%

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Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics

Chen¹,

Frid²

2003

Communications in Mathematical Physics

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103

142

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Abstract. A class of extended vector fields, called extended divergence-measure fields, is analyzed. These fields include vector fields in L p and vector-valued Radon measures, whose divergences are Radon measures. Such extended vector fields arise naturally in the study of the behavior of entropy solutions to the Euler equations for gas dynamics and other nonlinear systems of conservation laws. A new notion of normal traces over Lipschitz deformable surfaces is developed under which a generalized Gauss-Green theorem is established even for these extended fields. An explicit formula is obtained to calculate the normal traces over any Lipschitz deformable surface, suitable for applications, by using the neighborhood information of the fields near the surface and the level set function of the Lipschitz deformation surfaces. As an application, we prove the uniqueness and stability of Riemann solutions that may contain vacuum in the class of entropy solutions of the Euler equations for gas dynamics.

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Section: Example 12mentioning

confidence: 99%

Section: Example 12mentioning

confidence: 99%

Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics

Chen¹,

Frid²

2003

Communications in Mathematical Physics

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103

142

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“…Similar stability results have also been established for the Euler equations for isentropic gas dynamics in Chen [4], which are the limiting system of the relativistic Euler equations as the light speed tends to infinity; also see [8] for the nonisentropic case. We also refer the reader to Dafermos [14] for the stability of Lipschitz solutions for hyperbolic systems of conservation laws.…”

Section: Remark 12mentioning

confidence: 53%

Relativistic Euler equations for isentropic fluids: Stability of Riemann solutions with large oscillation

Chen¹,

Li²

2004

Z. angew. Math. Phys.

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We analyze global entropy solutions of the 2 × 2 relativistic Euler equations for isentropic fluids in special relativity and establish the uniqueness of Riemann solutions in the class of entropy solutions in L ∞ ∩ BV loc with arbitrarily large oscillation. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions implies their inviscid time-asymptotic stability under arbitrarily large L 1 ∩ L ∞ ∩ BV loc perturbation of the Riemann initial data, provided that the corresponding solutions are in L ∞ and have local bounded total variation that allows the linear growth in time. This approach is also extended to deal with the stability of Riemann solutions containing vacuum in the class of entropy solutions in L ∞ with arbitrarily large oscillation.

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“…It has been applied in various directions, e.g. for asymptotic stability problems in conservation laws [6,7], for relaxation or kinetic limits [1,38], or for comparing entropic measure-valued solutions and strong solutions for conservation laws [3].…”

Section: The Second Law Of Thermodynamics and Stabilitymentioning

confidence: 99%

Remarks on the Contributions of Constantine M. Dafermos to the Subject of Conservation Laws

Tzavaras

2012

Acta Mathematica Scientia

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Uniqueness and Stability of Riemann Solutions¶with Large Oscillation in Gas Dynamics

Cited by 65 publications

References 29 publications

Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics

Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics

Relativistic Euler equations for isentropic fluids: Stability of Riemann solutions with large oscillation

Remarks on the Contributions of Constantine M. Dafermos to the Subject of Conservation Laws

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