Abstract
We consider the Riemann problem for a system of mixed type that defines two hyperbolic phases satisfying general genuinely non linear hypothesis. We describe here all the global Riemann solvers that are continuous for theL ldistance with respect to initial data while conserving the natural symmetry properties of the system and coincinding with the Lax solution when defined.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Aberayatne and J. KnowlesKinetic relations and the propagations of phase boundaries in solidArch. Rational Mech. Anal., 114 (1991), 345–372.
J. Ball and James R.D.Fine phase mixtures as minimizers of energyArch. Rational Mech. Anal., 100 (1986), 13–52.
A. BressanGlobal solutions to systems of conservation laws by wave front trackingJ. Math. Anal. Appl.170 (1992), 414–432.
A. Bressan, G. Crasta and B. PiccoliWell posedness of the Cauchy problem for n x n systems of conservation lawsMemoirs of AMS, 145/694 (2000).
A. Bressan, Liu T.-P., Yang T.L 1 stability for systems of conservation laws, Arch.Rat. Mech.Anal.149(1999), 1–22.
R.M. Colombo and A. CorliContinuous dependence in conservation laws with phase transitionsSIAM J. Math. Anal.31 (1999), 34–62.
R.M. Colombo and H. FreistuehlerThe Riemann problem for two-phase media,preprint, RWTH - Aachen.
C.M. Dafermos, Hyperbolic systems of conservation laws, in “Systems of Nonlinear Partial Differential Equations”, J.M. Ball ed., NATO-ASI Series C 111 (1983), 25–70.
J. GlimmContinuous Dependance in Solutions in the large for hyperbolic systems of conservations lawsComm Pure and Appl. Math.101 (1965), 177–188.
H. Fan and M.Slemrod, The Riemann problem for systems of conservation laws of mixed type, in Shock induced transitions and phase structures in general media, IMA Vol. Math. Appl. 52Springer (New York)199361–91.
P. LeFlochPropagating phase boundaries: Formulation of the problem and existence via the Glimm methodArch. Rational Mech. Anal.123 (1993), 153–197.
J.M. Mercier and B. PiccoliGlobal continuous Riemann solver for non linear elasticityArch. Rat. Mech. Anal.156 (2001), 89–119
J.M. Mercier and B. Piccoli, Admissible Riemann solvers for genuinely nonlinear p-sytems of mixed typesubmitted to J. Differential Equations.
T.P. LiuThe Riemann problem for general systems of conservation lawsJ. Differential equations18 (1975), 218–234.
M. ShearerThe Riemann problem for a class of conservation laws of mixed type J. of Differential Equations46 (1982), 426–443.
M. Shearer and Y. YangThe Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearityProc. R. Soc. Edinb. Sect. A125 (1995), 675–699.
L. Truskinowsky, Kinks versus shocks in Shock induced transitions and phase structures in general media IMA Vol. Math. Appl.52, Springer (New York) 1993 185–229.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Mercier, JM., Piccoli, B. (2001). The Riemann Problem for Nonlinear Elasticity. In: Freistühler, H., Warnecke, G. (eds) Hyperbolic Problems: Theory, Numerics, Applications. ISNM International Series of Numerical Mathematics, vol 141. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8372-6_25
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8372-6_25
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9538-5
Online ISBN: 978-3-0348-8372-6
eBook Packages: Springer Book Archive