L 2 -type contraction for systems of conservation laws
Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 1-28.

The semi-group associated with the Cauchy problem for a scalar conservation law is known to be a contraction in L 1 . However it is not a contraction in L p for any p>1. Leger showed in [20] that for a convex flux, it is however a contraction in L 2 up to a suitable shift. We investigate in this paper whether such a contraction may happen for systems. The method is based on the relative entropy method. Our general analysis leads us to the new geometrical notion of Genuinely non-Temple systems. We treat in details two examples: – the Keyfitz–Kranzer system with rotationally invariant flux, for which the L 2 contraction holds true, – the Euler system of gas dynamics, for which it does not.

On sait que le semi-groupe associé au Problème de Cauchy pour une loi de conservation scalaire est contractant dans L 1 , mais qu’il ne l’est pas dans L p si p>1. Leger a montré dans [20], pour un flux convexe, une propriété de contraction dans L 2 moyennant une translation. Nous examinons ici la possibilité d’une telle propriété pour les systèmes. Notre analyse nous conduit à la notion géométrique de système Vraiment pas Temple. Nous traitons en détail deux exemples : – le système de Keyfitz et Kranzer avec flux isotrope, pour lequel la contraction a lieu, – le système de la dynamique des gaz, où ce n’est pas le cas.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.1
Classification: 35L65, 35L67, 35L40
Keywords: Conservation laws, relative entropy, shock stability, Temple systems
Mot clés : Lois de conservation, entropie relative, stabilité des ondes de choc, systèmes de Temple

Denis Serre 1; Alexis F. Vasseur 2

1 UMPA, ENS-Lyon 46 allée d’Italie, 69364 Lyon Cedex 07, France
2 University of Texas at Austin 1 University Station C1200, Austin, TX 78712-0257, USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Denis Serre; Alexis F. Vasseur. $L^2$-type contraction for systems of conservation laws. Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 1-28. doi : 10.5802/jep.1. https://jep.centre-mersenne.org/articles/10.5802/jep.1/

[1] C. Bardos, F. Golse & C. D. Levermore - “Fluid dynamic limits of kinetic equations. I. Formal derivations”, J. Statist. Phys. 63 (1991) no. 1-2, p. 323-344 | DOI | MR

[2] C. Bardos, F. Golse & C. D. Levermore - “Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation”, Comm. Pure Appl. Math. 46 (1993) no. 5, p. 667-753 | DOI | MR | Zbl

[3] B. Barker, H. Freistühler & K. Zumbrun - “Convex entropy, Hopf bifurcation, and viscous and inviscid shock stability” (2013), Preprint | Zbl

[4] F. Berthelin, A. E. Tzavaras & A. Vasseur - “From discrete velocity Boltzmann equations to gas dynamics before shocks”, J. Statist. Phys. 135 (2009) no. 1, p. 153-173 | DOI | MR | Zbl

[5] F. Berthelin & A. Vasseur - “From kinetic equations to multidimensional isentropic gas dynamics before shocks”, SIAM J. Math. Anal. 36 (2005) no. 6, p. 1807-1835 | DOI | MR | Zbl

[6] A. Bressan - Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem, Oxford University Press, Oxford, 2000 | Zbl

[7] G.-Q. Chen - “Vacuum states and global stability of rarefaction waves for compressible flow”, Methods Appl. Anal. 7 (2000), p. 337-361 | MR | Zbl

[8] G.-Q. Chen & H. Frid - “Large-time behavior of entropy solutions of conservation laws”, J. Differential Equations 152 (1999) no. 2, p. 308-357 | DOI | MR | Zbl

[9] G.-Q. Chen & H. Frid - “Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations”, Trans. Amer. Math. Soc. 353 (2001) no. 3, p. 1103-1117 | DOI | MR

[10] G.-Q. Chen, H. Frid & Y. Li - “Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics”, Comm. Math. Phys. 228 (2002) no. 2, p. 201-217 | DOI | MR

[11] G.-Q. Chen & Y. Li - “Stability of Riemann solutions with large oscillation for the relativistic Euler equations”, J. Differential Equations 202 (2004) no. 2, p. 332-353 | DOI | MR

[12] C. M. Dafermos - “The second law of thermodynamics and stability”, Arch. Rational Mech. Anal. 70 (1979) no. 2, p. 167-179 | DOI | MR | Zbl

[13] C. M. Dafermos - “Entropy and the stability of classical solutions of hyperbolic systems of conservation laws”, in Recent mathematical methods in nonlinear wave propagation (Montecatini Terme, 1994), Lecture Notes in Math., vol. 1640, Springer, Berlin, 1996, p. 48-69 | DOI | MR | Zbl

[14] C. M. Dafermos - Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, vol. 325, Springer-Verlag, Berlin, 2000 | MR | Zbl

[15] R. J. DiPerna - “Uniqueness of solutions to hyperbolic conservation laws”, Indiana Univ. Math. J. 28 (1979) no. 1, p. 137-188 | DOI | MR | Zbl

[16] F. Golse & L. Saint-Raymond - “The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels”, Invent. Math. 155 (2004) no. 1, p. 81-161 | DOI | MR | Zbl

[17] B. L. Keyfitz & H. C. Kranzer - “A system of nonstrictly hyperbolic conservation laws arising in elasticity theory”, Arch. Rational Mech. Anal. 72 (1979/80) no. 3, p. 219-241 | DOI | MR

[18] Y.-S. Kwon & A. Vasseur - “Strong traces for solutions to scalar conservation laws with general flux”, Arch. Rational Mech. Anal. 185 (2007) no. 3, p. 495-513 | DOI | MR | Zbl

[19] P. D. Lax - “Hyperbolic systems of conservation laws. II”, Comm. Pure Appl. Math. 10 (1957), p. 537-566 | DOI | MR | Zbl

[20] N. Leger - “L 2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method”, Arch. Rational Mech. Anal. 199 (2011), p. 761-778 | DOI | MR | Zbl

[21] N. Leger & A. Vasseur - “Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations”, Arch. Rational Mech. Anal. 201 (2011) no. 1, p. 271-302 | DOI | MR | Zbl

[22] P.-L. Lions & N. Masmoudi - “From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II”, Arch. Rational Mech. Anal. 158 (2001) no. 3, p. 173-193, 195–211 | DOI | MR | Zbl

[23] N. Masmoudi & L. Saint-Raymond - “From the Boltzmann equation to the Stokes–Fourier system in a bounded domain”, Comm. Pure Appl. Math. 56 (2003) no. 9, p. 1263-1293 | DOI | MR | Zbl

[24] A. Mellet & A. Vasseur - “Asymptotic analysis for a Vlasov–Fokker–Planck/compressible Navier–Stokes system of equations”, Comm. Math. Phys. 281 (2008) no. 3, p. 573-596 | DOI | MR | Zbl

[25] L. Saint-Raymond - “Convergence of solutions to the Boltzmann equation in the incompressible Euler limit”, Arch. Rational Mech. Anal. 166 (2003) no. 1, p. 47-80 | DOI | MR | Zbl

[26] L. Saint-Raymond - “From the BGK model to the Navier-Stokes equations”, Ann. Sci. École Norm. Sup. (4) 36 (2003) no. 2, p. 271-317 | DOI | Numdam | MR | Zbl

[27] D. Serre - “Oscillations non linéaires des systèmes hyperboliques: méthodes et résultats qualitatifs”, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) no. 3-4, p. 351-417 | DOI | Numdam | Zbl

[28] D. Serre - Systems of conservation laws II, Cambridge University Press, Cambridge, 2000

[29] B. Texier & K. Zumbrun - “Entropy criteria and stability of extreme shocks: a remark on a paper of Leger and Vasseur” (2013), Preprint | Zbl

[30] A. E. Tzavaras - “Relative entropy in hyperbolic relaxation”, Commun. Math. Sci. 3 (2005) no. 2, p. 119-132 | DOI | MR | Zbl

[31] A. Vasseur - “Recent results on hydrodynamic limits”, in Handbook of differential equations: evolutionary equations. Vol. IV (C. Dafermos & M. Pokorny, eds.), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, p. 323-376 | Zbl

[32] D. Wagner - “Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions”, J. Differential Equations 68 (1987) no. 1, p. 118-136 | DOI | MR

[33] H.-T. Yau - “Relative entropy and hydrodynamics of Ginzburg–Landau models”, Lett. Math. Phys. 22 (1991) no. 1, p. 63-80 | DOI | MR | Zbl

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