Stability of the Riemann semigroup with respect to the kinetic condition
Authors:
Rinaldo M. Colombo and Andrea Corli
Journal:
Quart. Appl. Math. 62 (2004), 541-551
MSC:
Primary 35L65; Secondary 47H20, 76T99
DOI:
https://doi.org/10.1090/qam/2086045
MathSciNet review:
MR2086045
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Abstract: This note deals with systems of hyperbolic conservation laws that are endowed with a generalized kinetic relation and develop phase transitions. The ${L^1}$-Lipschitzean continuous dependence of the solution from the kinetic relation is proved. Preliminarily, we rephrase several results known in the case of standard conservation laws to the case comprising phase boundaries.
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- Rohan Abeyaratne and James K. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal. 114 (1991), no. 2, 119–154. MR 1094433, DOI https://doi.org/10.1007/BF00375400
- Stefano Bianchini, On the shift differentiability of the flow generated by a hyperbolic system of conservation laws, Discrete Contin. Dynam. Systems 6 (2000), no. 2, 329–350. MR 1739381, DOI https://doi.org/10.3934/dcds.2000.6.329
- Stefano Bianchini and Rinaldo M. Colombo, On the stability of the standard Riemann semigroup, Proc. Amer. Math. Soc. 130 (2002), no. 7, 1961–1973. MR 1896028, DOI https://doi.org/10.1090/S0002-9939-02-06568-1
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- Rinaldo M. Colombo and Andrea Corli, Continuous dependence in conservation laws with phase transitions, SIAM J. Math. Anal. 31 (1999), no. 1, 34–62. MR 1720130, DOI https://doi.org/10.1137/S0036141097331871
- Rinaldo M. Colombo and Andrea Corli, Sonic hyperbolic phase transitions and Chapman-Jouguet detonations, J. Differential Equations 184 (2002), no. 2, 321–347. MR 1929881, DOI https://doi.org/10.1006/jdeq.2001.4131
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R. Abeyaratne and J. K. Knowles. Implications of viscosity and strain-gradient effects for the kinetics of propagating phase boundaries in solids. SIAM J. Appl. Math., 51(5):1205–1221, 1991.
R. Abeyaratne and J. K. Knowles. Kinetic relations and the propagation of phase boundaries in solids. Arch. Rational Mech. Anal., 114(2): 119–154, 1991.
S. Bianchini. On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete Contin. Dynam. Systems, 6(2):329–350, 2000.
S. Bianchini and R. M. Colombo. On the stability of the standard Riemann semigroup. Proc. Amer. Math. Soc., 130(7): 1961–1973 (electronic), 2002.
A. Bressan. Hyperbolic systems of conservation laws. Oxford University Press, Oxford, 2000.
A. Bressan and R. M. Colombo. Unique solutions of $2 \times 2$ conservation laws with large data. Indiana Univ. Math. J., 44(3):677–725, 1995.
R. M. Colombo and A. Corli. Continuous dependence in conservation laws with phase transitions. SIAM Journal Math. Anal., 31(1):34–62, 1999.
R. M. Colombo and A. Corli. Sonic hyperbolic phase transitions and Chapman-Jouguet detonations. J. Differential Equations, 184(2):321–347, 2002.
R. M. Colombo and A. Corli. Sonic and kinetic phase transitions with applications to Chapman-Jouguet deflagrations. NoDEA Nonlinear Differential Equations Appl., to appear.
A. Corli. Noncharacteristic phase boundaries for general systems of conservation laws. Ital. J. Pure Appl. Math., 6:43–62 (2000), 1999.
C. M. Dafermos. Hyperbolic conservation laws in continuum physics. Springer-Verlag, New York, first edition, 2000.
W. Fickett and W. C. Davis. Detonation. University of California Press, Berkeley, 1979.
H. Freistühler. A short note on the persistence of ideal shock waves. Arch. Math. (Basel), 64(4) :344–352, 1995.
E. Godlewski and P.-A. Raviart. Numerical approximation of hyperbolic systems of conservation laws, volume 118 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996.
J. Hu and P. G. LeFloch. ${L^1}$ continuous dependence property for systems of conservation laws. Arch. Ration. Mech. Anal., 151(1):45–93, 2000.
P. D. Lax. Hyperbolic systems of conservation laws. II. Comm. Pure Appl. Math., 10:537–566, 1957.
M. Lewicka. ${L^1}$ stability of patterns of non-interacting large shock waves. Indiana Univ. Math. J., 49(4):1515–1537, 2000.
M. Lewicka and K. Trivisa. On the ${L^1}$ well posedness of systems of conservation laws near solutions containing two large shocks. J. Differential Equations, 179(1):133–177, 2002.
M. Slemrod. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal., 81(4):301–315, 1983.
M. Slemrod. A limiting “viscosity” approach to the Riemann problem for materials exhibiting change of phase. Arch. Rational Mech. Anal., 105(4):327–365, 1989.
Z. H. Teng, A. J. Chorin, and T. P. Liu. Riemann problems for reacting gas, with applications to transition. SIAM J. Appl. Math., 42(5):964–981, 1982.
L. Truskinovsky. About the “normal growth” approximation in the dynamical theory of phase transitions. Contin. Mech. Thermodyn., 6(3): 185–208, 1994.
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