Linear stability of shock profiles for a rate-type viscoelastic system with relaxation
Authors:
Tao Luo and Denis Serre
Journal:
Quart. Appl. Math. 56 (1998), 569-586
MSC:
Primary 35L67; Secondary 35Q72, 73F15
DOI:
https://doi.org/10.1090/qam/1632322
MathSciNet review:
MR1632322
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Abstract: We investigate the linear stability of a shock profile for a rate-type viscoelastic system with relaxation. The linear stability of shock profiles for a general initial perturbation is proved by introducing new waves through time-asymptotic expansion and using energy estimates.
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C. Cercignani, The Boltzmann Equation and its Applications, Springer-Verlag, New York, 1998
G. Q. Chen, C. D. Levermore, and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47, 787–830 (1994)
R. E. Caflisch and G. C. Papanicolaou, The fluid dynamical limit of a nonlinear model Boltzmann equation, Comm. Pure Appl. Math. 32, 589–616 (1979)
J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95, 325–344 (1986)
G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, 1934
L. Hsiao and T. Luo, The stability of travelling wave solutions for a rate type viscoelastic system, to appear
K. Inoue and T. Nishida, On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas, Appl. Math. Optimization 3, 27–49 (1976)
S. Kawashima and A. Matsumura, Asymptotic stability to travelling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101, 92–127 (1985)
T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108, 153–175 (1987)
T. P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Memoirs Amer. Math. Soc. 328 (1986)
A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 2, 17–25 (1985)
R. Natalini, Convergence to the equilibrium solutions for the system of conservation laws with relaxation, preprint
T. Platkowski and R. Illner, Discrete velocity models of the Boltzmann equation: a survey of the mathematical aspects of the theory, SIAM Review 30, 213–255 (1988)
M. Slemlod and A. E. Tzavaras, Self-similar fluid-dynamic limits for the Broadwell system, Arch. Rational Mech. Anal. 122, 353–392 (1993)
I. Suliciu, On modelling phase transitions by means of rate-type constitutive equations. Shock wave structure, Internat. J. Engrg. Sci. 28, 829–841 (1990)
A. Szepessy and Z. Xin, Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal. 122, 53–103 (1993)
Z. Xin, The fluid-dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks, Comm. Pure Appl. Math. 44, 679–713 (1991)
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© Copyright 1998
American Mathematical Society