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August  2016, 36(8): 4569-4577. doi: 10.3934/dcds.2016.36.4569

The relative entropy method for the stability of intermediate shock waves; the rich case

Denis Serre 1, and  Alexis F. Vasseur 2,

1. 

UMPA, ENS-Lyon 46, allée d'Italie 69364 LYON Cedex 07, France
2. 

Department of Mathematics, University of Texas at Austin, 1 University Station – C1200, Austin, TX 78712-0257, United States

Received  April 2015 Revised  August 2015 Published  March 2016

M.-J. Kang and one of us [2] developed a new version of the relative entropy method, which is efficient in the study of the long-time stability of extreme shocks. When a system of conservation laws is rich, we show that this can be adapted to the case of intermediate shocks.
Keywords: Shock waves, stability, hyperbolic conservation laws, entropy method., rich systems.
Mathematics Subject Classification: 35L65, 35L6.
Citation: Denis Serre, Alexis F. Vasseur. The relative entropy method for the stability of intermediate shock waves; the rich case. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4569-4577. doi: 10.3934/dcds.2016.36.4569
References:
[1]

S. Benzoni-Gavage, D. Serre and K. Zumbrun, Alternate Evans functions and viscous shock waves, SIAM J. Math. Anal., 32 (2001), 929-962. doi: 10.1137/S0036141099361834.  Google Scholar

[2]

M.-J. Kang and A. Vasseur, Criteria on contractions for entropic discontinuities of systems of conservation laws, reprint, arXiv:1505.02245 (2015). Google Scholar

[3]

P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[4]

N. Leger, $L^2$-stability estimates for shock solutions of scalar conservation laws using the relative entropy method, Arch. Rational Mech. Anal., 199 (2011), 761-778. doi: 10.1007/s00205-010-0341-7.  Google Scholar

[5]

N. Leger and A. Vasseur, Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations, Arch. Ration. Mech. Anal., 201 (2011), 271-302. doi: 10.1007/s00205-011-0431-1.  Google Scholar

[6]

T.-P. Liu, Pointwise convergence to $N$-waves for solutions of hyperbolic conservation laws, Bull. Inst. Math. Acad. Sinica, 15 (1987), 1-17.  Google Scholar

[7]

A. Majda, The stability of multidimensional shock fronts, Memoirs Amer. Math. Soc., 41 (1983), iv+95 pp. doi: 10.1090/memo/0275.  Google Scholar

[8]

A. Majda, The existence of multidimensional shock fronts, Memoirs Amer. Math. Soc., 43 (1983), v+93 pp. doi: 10.1090/memo/0281.  Google Scholar

[9]

D. Serre, Richness and the classification of quasilinear hyperbolic systems. Multidimensional hyperbolic problems and computations (Minneapolis, MN, 1989), IMA Vol. Math. Appl., 29, Springer, New York, (1991), 315-333. doi: 10.1007/978-1-4613-9121-0_24.  Google Scholar

[10]

D. Serre, Systems of Conservation Laws, II, Cambridge Univ. Press, 2000.  Google Scholar

[11]

D. Serre, The structure of dissipative viscous system of conservation laws, PhysicaD, 239 (2010), 1381-1386. doi: 10.1016/j.physd.2009.03.014.  Google Scholar

[12]

D. Serre, Long-time stability in systems of conservation laws, using relative entropy/energy, Arch. Ration. Mech. Anal., 219 (2016), 679-699. doi: 10.1007/s00205-015-0903-9.  Google Scholar

[13]

D. Serre, A. Vasseur, $L^2$-type contraction for systems of conservation laws, Journal de l'École polytechnique; Mathématiques, 1 (2014), 1-28. doi: 10.5802/jep.1.  Google Scholar

show all references

References:
[1]

S. Benzoni-Gavage, D. Serre and K. Zumbrun, Alternate Evans functions and viscous shock waves, SIAM J. Math. Anal., 32 (2001), 929-962. doi: 10.1137/S0036141099361834.  Google Scholar

[2]

M.-J. Kang and A. Vasseur, Criteria on contractions for entropic discontinuities of systems of conservation laws, reprint, arXiv:1505.02245 (2015). Google Scholar

[3]

P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[4]

N. Leger, $L^2$-stability estimates for shock solutions of scalar conservation laws using the relative entropy method, Arch. Rational Mech. Anal., 199 (2011), 761-778. doi: 10.1007/s00205-010-0341-7.  Google Scholar

[5]

N. Leger and A. Vasseur, Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations, Arch. Ration. Mech. Anal., 201 (2011), 271-302. doi: 10.1007/s00205-011-0431-1.  Google Scholar

[6]

T.-P. Liu, Pointwise convergence to $N$-waves for solutions of hyperbolic conservation laws, Bull. Inst. Math. Acad. Sinica, 15 (1987), 1-17.  Google Scholar

[7]

A. Majda, The stability of multidimensional shock fronts, Memoirs Amer. Math. Soc., 41 (1983), iv+95 pp. doi: 10.1090/memo/0275.  Google Scholar

[8]

A. Majda, The existence of multidimensional shock fronts, Memoirs Amer. Math. Soc., 43 (1983), v+93 pp. doi: 10.1090/memo/0281.  Google Scholar

[9]

D. Serre, Richness and the classification of quasilinear hyperbolic systems. Multidimensional hyperbolic problems and computations (Minneapolis, MN, 1989), IMA Vol. Math. Appl., 29, Springer, New York, (1991), 315-333. doi: 10.1007/978-1-4613-9121-0_24.  Google Scholar

[10]

D. Serre, Systems of Conservation Laws, II, Cambridge Univ. Press, 2000.  Google Scholar

[11]

D. Serre, The structure of dissipative viscous system of conservation laws, PhysicaD, 239 (2010), 1381-1386. doi: 10.1016/j.physd.2009.03.014.  Google Scholar

[12]

D. Serre, Long-time stability in systems of conservation laws, using relative entropy/energy, Arch. Ration. Mech. Anal., 219 (2016), 679-699. doi: 10.1007/s00205-015-0903-9.  Google Scholar

[13]

D. Serre, A. Vasseur, $L^2$-type contraction for systems of conservation laws, Journal de l'École polytechnique; Mathématiques, 1 (2014), 1-28. doi: 10.5802/jep.1.  Google Scholar

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