Stability of Jin-Xin relaxation shocks

Humpherys, Jeffrey

doi:10.1090/qam/1976368

Author: Jeffrey Humpherys
Journal: Quart. Appl. Math. 61 (2003), 251-263
MSC: Primary 35L60; Secondary 35B35, 35L67
DOI: https://doi.org/10.1090/qam/1976368
MathSciNet review: MR1976368
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Abstract: We examine the spectrum of shock profiles for the Jin-Xin relaxation scheme for systems of hyperbolic conservation laws in one spatial dimension. By using a weighted norm estimate, we prove that these shock profiles exhibit strong spectral stability in the weak shock limit.

Leon Q. Brin, Numerical testing of the stability of viscous shock waves, Math. Comp. 70 (2001), no. 235, 1071–1088. MR 1710652, DOI https://doi.org/10.1090/S0025-5718-00-01237-0

P. Godillon, Stabilité linéaire des profils pour les systémes avec relaxation semi-linéaire, Phys. D. 148 (2001), no. 3-4, 289–316

Jonathan Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325–344. MR 853782, DOI https://doi.org/10.1007/BF00276840
Jeffrey Humpherys and Kevin Zumbrun, Spectral stability of small-amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems, Z. Angew. Math. Phys. 53 (2002), no. 1, 20–34. MR 1889177, DOI https://doi.org/10.1007/s00033-002-8139-6
Shi Jin and Zhou Ping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), no. 3, 235–276. MR 1322811, DOI https://doi.org/10.1002/cpa.3160480303
Hailiang Liu, Asymptotic stability of relaxation shock profiles for hyperbolic conservation laws, J. Differential Equations 192 (2003), no. 2, 285–307. MR 1990842, DOI https://doi.org/10.1016/S0022-0396%2803%2900124-4
Andrew Majda and Robert L. Pego, Stable viscosity matrices for systems of conservation laws, J. Differential Equations 56 (1985), no. 2, 229–262. MR 774165, DOI https://doi.org/10.1016/0022-0396%2885%2990107-X

C. Mascia and K. Zumbrun, Pointwise Green’s bounds and stability of relaxation shocks, in preparation

Yanni Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal. 150 (1999), no. 3, 225–279. MR 1738119, DOI https://doi.org/10.1007/s002050050188

K. Zumbrun, Stability of multidimensional shock fronts, Lecture Notes, TMR Summer School, Köchel, May 1999

Kevin Zumbrun and Peter Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (1998), no. 3, 741–871. MR 1665788, DOI https://doi.org/10.1512/iumj.1998.47.1604