×

Linear stability of shock profiles for systems of conservation laws with semi-linear relaxation. (English) Zbl 1076.76033

Summary: The Evans function theory, which has recently been applied to the study of linear stability of viscous shock profiles, is developed below for semi-linear relaxation. We study the linear stability of shock profiles in the Lax undercompressive and overcompressive cases. The results we obtain are similar to those found for viscous approximations by R. A. Gardner and K. Zumbrun [Commun. Pure Appl. Math. 51, No. 7, 797–855 (1998; Zbl 0933.35136)].

MSC:

76E17 Interfacial stability and instability in hydrodynamic stability
76L05 Shock waves and blast waves in fluid mechanics

Citations:

Zbl 0933.35136
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. Benzoni-Gavage, Stability of semi-discrete shock profiles by means of an Evans function in infinite dimensions, Preprint, ENS Lyon, 1999.; S. Benzoni-Gavage, Stability of semi-discrete shock profiles by means of an Evans function in infinite dimensions, Preprint, ENS Lyon, 1999. · Zbl 0937.35181
[2] S. Benzoni-Gavage, D. Serre, K. Zumbrun, Alternate Evans functions and viscous shock waves, Preprint, ENS Lyon, 1999.; S. Benzoni-Gavage, D. Serre, K. Zumbrun, Alternate Evans functions and viscous shock waves, Preprint, ENS Lyon, 1999. · Zbl 0985.34075
[3] S. Bianchini, On a Glimm type functional for a special Jin-Xin relaxation model, Preprint, 1999.; S. Bianchini, On a Glimm type functional for a special Jin-Xin relaxation model, Preprint, 1999. · Zbl 0981.35037
[4] Chen, G. Q.; Levermore, C. D.; Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math., 47, 6, 787-830 (1994) · Zbl 0806.35112
[5] Chen, G. Q.; Liu, T.-P., Zero relaxation and dissipation limits for hyperbolic conservation laws, Commun. Pure Appl. Math., 46, 5, 755-781 (1993) · Zbl 0797.35113
[6] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2000.; C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2000. · Zbl 0940.35002
[7] DiPerna, R. J., Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal., 82, 1, 27-70 (1983) · Zbl 0519.35054
[8] Evans, J. W., Nerve axon equations. I. Linear approximations, Indiana Univ. Math. J., 21, 877-885 (1971/72) · Zbl 0235.92002
[9] Foy, L. R., Steady state solutions of hyperbolic systems of conservation laws with viscosity terms, Commun. Pure Appl. Math., 17, 177-188 (1964) · Zbl 0178.11902
[10] Freistühler, H.; Liu, T.-P., Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws, Commun. Math. Phys., 153, 1, 147-158 (1993) · Zbl 0768.76024
[11] Freistühler, H.; Szmolyan, P., Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal., 26, 1, 112-128 (1995) · Zbl 0817.34028
[12] H. Freistühler, K. Zumbrun, Examples of unstable viscous shock waves, Technical Report, Institut für Mathematik, 1998.; H. Freistühler, K. Zumbrun, Examples of unstable viscous shock waves, Technical Report, Institut für Mathematik, 1998.
[13] Gardner, R. A.; Zumbrun, K., The Gap lemma and geometric criteria for instability of viscous shock profiles, Commun. Pure Appl. Math., 51, 7, 797-855 (1998)
[14] Gilbarg, D., The existence and limit behavior of the one-dimensional shock layer, Am. J. Math., 73, 256-274 (1951) · Zbl 0044.21504
[15] Goodman, J., Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal., 95, 4, 325-344 (1986) · Zbl 0631.35058
[16] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer, Berlin, 1981.; D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer, Berlin, 1981. · Zbl 0456.35001
[17] Jin, S.; Xin, Z. P., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math., 48, 3, 235-276 (1995) · Zbl 0826.65078
[18] Jones, C. K.R. T., Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans. Am. Math. Soc., 286, 2, 431-469 (1984) · Zbl 0567.35044
[19] H. Liu, G. Warnecke, Convergence rates for relaxation schemes approximating conservation laws, Preprint, 1999.; H. Liu, G. Warnecke, Convergence rates for relaxation schemes approximating conservation laws, Preprint, 1999. · Zbl 0954.35108
[20] Liu, T.-P., Hyperbolic conservation laws with relaxation, Commun. Math. Phys., 108, 1, 153-175 (1987) · Zbl 0633.35049
[21] T.-P. Liu, On the viscosity criterion for hyperbolic conservation laws, in: Viscous Profiles and Numerical Methods for Shock Waves (Raleigh, NC, 1990), SIAM, Philadelphia, PA, 1991, pp. 105-114.; T.-P. Liu, On the viscosity criterion for hyperbolic conservation laws, in: Viscous Profiles and Numerical Methods for Shock Waves (Raleigh, NC, 1990), SIAM, Philadelphia, PA, 1991, pp. 105-114.
[22] T.-P. Liu, Nonlinear stability and instability of overcompressive shock waves, in: Shock Induced Transitions and Phase Structures in General Media, IMA Math. Appl. Vol. 52. Springer, New York, 1993, pp. 159-167.; T.-P. Liu, Nonlinear stability and instability of overcompressive shock waves, in: Shock Induced Transitions and Phase Structures in General Media, IMA Math. Appl. Vol. 52. Springer, New York, 1993, pp. 159-167.
[23] T.-P. Liu, Z. Xin, Overcompressive shock waves, in: Nonlinear Evolution Equations that Change Type, IMA Math. Appl. Vol. 27. Springer, New York, 1990, pp. 139-145.; T.-P. Liu, Z. Xin, Overcompressive shock waves, in: Nonlinear Evolution Equations that Change Type, IMA Math. Appl. Vol. 27. Springer, New York, 1990, pp. 139-145.
[24] Majda, A.; Pego, R. L., Stable viscosity matrices for systems of conservation laws, J. Differ. Equations, 56, 2, 229-262 (1985) · Zbl 0512.76067
[25] Mallet-Paret, J., The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Differ. Equations, 11, 1, 1-47 (1999) · Zbl 0927.34049
[26] Mock, M. S., A topological degree for orbits connecting critical points of autonomous systems, J. Differ. Equations, 38, 2, 176-191 (1980) · Zbl 0417.34053
[27] R. Natalini, Recent results on hyperbolic relaxation problems, in: Analysis of Systems of Conservation Laws (Aachen, 1997), Monogr. Surv. Pure Appl. Math., Vol. 99. Chapman and Hall, CRC, Boca Raton, FL, 1999, pp. 128-198.; R. Natalini, Recent results on hyperbolic relaxation problems, in: Analysis of Systems of Conservation Laws (Aachen, 1997), Monogr. Surv. Pure Appl. Math., Vol. 99. Chapman and Hall, CRC, Boca Raton, FL, 1999, pp. 128-198. · Zbl 0940.35127
[28] S. Schecter, M. Shearer, Transversality for undercompressive shocks in Riemann problems, in: Viscous Profiles and Numerical Methods for Shock Waves (Raleigh, NC, 1990), SIAM, Philadelphia, PA, 1991, pp. 142-154.; S. Schecter, M. Shearer, Transversality for undercompressive shocks in Riemann problems, in: Viscous Profiles and Numerical Methods for Shock Waves (Raleigh, NC, 1990), SIAM, Philadelphia, PA, 1991, pp. 142-154. · Zbl 0752.35035
[29] D. Serre, Systems of Conservation Laws 1, Cambridge University Press, Cambridge, 1999. Hyperbolicity, entropies, shock waves (Translated from the 1996 French original by I.N. Sneddon).; D. Serre, Systems of Conservation Laws 1, Cambridge University Press, Cambridge, 1999. Hyperbolicity, entropies, shock waves (Translated from the 1996 French original by I.N. Sneddon). · Zbl 0930.35001
[30] Serre, D., Relaxations semi-linéaire et cinétique des systèmes de lois de conservation, Ann. IHP Anal. Non-linéaire, 17, 169-192 (2000) · Zbl 0963.35117
[31] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd Edition, Springer, New York, 1994.; J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd Edition, Springer, New York, 1994. · Zbl 0807.35002
[32] G.B. Whitham, Linear and Nonlinear Waves, Wiley/Interscience, New York, 1974.; G.B. Whitham, Linear and Nonlinear Waves, Wiley/Interscience, New York, 1974. · Zbl 0373.76001
[33] Yong, W.-A., Boundary conditions for hyperbolic systems with stiff source terms, Indiana Univ. Math. J., 48, 1, 115-137 (1999) · Zbl 0929.35080
[34] W.-A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differ. Equations 155 (1) 1999.; W.-A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differ. Equations 155 (1) 1999.
[35] Zumbrun, K.; Howard, P., Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47, 3, 741-871 (1998) · Zbl 0928.35018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.