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Long-time Stability in Systems of Conservation Laws, Using Relative Entropy/Energy

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Abstract

We study the long-time stability of shock-free solutions of hyperbolic systems of conservation laws, under an arbitrarily large initial disturbance in L 2∩ L ∞. We use the relative entropy method, a robust tool which allows us to consider rough and large disturbances. We display practical examples in several space dimensions, for scalar equations as well as isentropic gas dynamics. For full gas dynamics, we use a trick from Chen [1], in which the estimate is made in terms of the relative mechanical energy instead of the relative mathematical entropy.

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References

  1. Chen G.-Q.: Vacuum states and global stability of rarefaction waves for compressible flow. Methods Appl. Anal., 7, 337–361 (2000)

    MATH  MathSciNet  Google Scholar 

  2. Chen, G.-Q., Chen, J.: Stability of rarefaction waves and vacuum states for the multidimensional Euler equations. J. Hyperbolic Differential Equations, 4(1), 105–122 (2007)

  3. Chen, G.-Q., Frid, H.: Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations. Trans. Am. Math. Soc., 353, 1103–1117 (2001)

  4. Chen, G.-Q., Li, Y.: Stability of Riemann solutions with large oscillation for the relativistic Euler equations. J. Differ. Eq., 202(2), 332–353 (2004)

  5. Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Rational Mech. Anal., 70(2), 167–179 (1979)

  6. Dafermos, C.M.: Entropy and the stability of classical solutions of hyperbolic systems of conservation laws. In Recent mathematical methods in nonlinear wave propagation (Montecatini Terme, 1994), volume 1640 of Lecture Notes in Math., pages 48–69. Springer, Berlin (1996)

  7. Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, vol. 325. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2000)

  8. DiPerna R.J.: Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J., 28, 137–188 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  9. Grassin M.: Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J., 47, 1397–1432 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hoff, D.: The sharp form of Oleĭnik’s entropy condition in several space variables. Trans. Am. Math. Soc., 276, 707–714 (1983)

  11. Serre D.: Solutions classiques globales des équations d’Euler pour un fluide parfait compressible. Ann. Inst. Fourier, 47, 139–153 (1997)

    Article  MATH  Google Scholar 

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  1. UMPA, UMR CNRS–ENS Lyon # 5669, École Normale Supérieure de Lyon, 46, allée d’Italie, 69364, Lyon Cedex 07, France

    Denis Serre

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  1. Denis Serre
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Correspondence to Denis Serre.

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Communicated by C. Dafermos

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Serre, D. Long-time Stability in Systems of Conservation Laws, Using Relative Entropy/Energy. Arch Rational Mech Anal 219, 679–699 (2016). https://doi.org/10.1007/s00205-015-0903-9

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  • DOI: https://doi.org/10.1007/s00205-015-0903-9

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