Abstract
In this paper we validate the generalized geometric entropy criterion for admissibility of shocks in systems which change type. This condition states that a shock between a state in a hyperbolic region and one in a nonhyperbolic region is admissible if the Lax geometric entropy criterion, based on the number of characteristics entering the shock, holds, where now the real part of a complex characteristic replaces the characteristic speed itself. We test this criterion by a nonlinear inviscid perturbation. We prove that the perturbed Cauchy problem in the elliptic region has a solution for a uniform time if the data lie in a suitable class of analytic functions and show that under small perturbations of the data a perturbed shock and a perturbed solution in the hyperbolic region exist, also for a uniform time.
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Allen, M. B., III, Behie, G. A., and Trangenstein, J. A. (1988).Multiphase Flow in Porous Media: Mechanics, Mathematics, and Numerics, Lecture Notes in Engineering, Vol. 34, Springer-Verlag, New York.
Ames, K. A., and Keyfitz, B. L. (1991). Stability of shocks in systems that change type: the linear approximation. In Engquist, B., and Gustafsson, B. (eds.),Third International Conference on Hyperbolic Problems: Theory, Numerical Methods and Applications, Chartwell-Bratt-Studentlitteratur, Lund, pp. 36–47.
Bell, J. B., Trangenstein, J. A., and Shubin, G. R. (1986). Conservation laws of mixed type describing three-phase flow in porous media.SIAM J. Appl. Math. 46, 1000–1023.
Courant, R., and Lax, P.D. (1949). On nonlinear partial differential equations with two independent variables.Comm. Pure Appl. Math. 2, 255–273.
Garabedian, P. R. (1964).Partial Differential Equations, Wiley-Interscience, New York.
Göldner, S. (1949).Existence and Uniqueness Theorems for Two Systems of Non-Linear Hyperbolic Differential Equations for Functions of Two Independent Variables, Dissertation, New York University, New York.
Harabetian, E. (1986). A convergent series expansion for hyperbolic systems of conservation laws.Trans. Am. Math. Soc. 294, 383–424.
Hartman, P. (1964).Ordinary Differential Equations, Wiley, New York.
James, R. D. (1980). The propagation of phase boundaries in elastic bars.Arch. Rat. Mech. Anal. 73, 125–158.
John, F. (1971).Partial Differential Equations, 4th ed., Springer-Verlag, New York.
Keller, J. B., and Ting, L. (1966). Periodic vibrations of systems governed by nonlinear partial differential equations.Comm. Pure Appl. Math. XIX, 371–420.
Keyfitz, B. L. (1989). Change of type in three-phase flow: a simple analogue.J. Diff. Eq. 80, 280–305.
Keyfitz, B. L. (1990). Shocks near the sonic line: a comparison between steady and unsteady models for change of type. In Keyfitz, B. L., and Shearer, M. (eds.),Nonlinear Evolution Equations that Change Type, IMA Vol. 27, Springer, New York, pp. 89–106.
Keyfitz, B. L. (1991). Admissibility conditions for shocks in systems that change type.SIAM J. Math. An. 22, 1284–1292.
Keyfitz, B. L. (1992). Development of singularities in Riemann invariants. University of Houston Mathematics Department Preprint.
Li Ta-Tsien and Yu Wen-ci (1985).Boundary Value Problems for Quasilinear Hyperbolic Systems, Mathematics Department, Duke University, Durham, NC.
Majda, A. J. (1983). The stability of multi-dimensional shock fronts.Am. Math. Soc. Mem. 275.
Narasimhan, R. (1971).Several Complex Variables, University of Chicago Press, Chicago.
Slemrod, M. (1983). Admissibility criteria for propagating phase boundaries in a van der Waals fluid.Arch. Rat. Mech. Anal. 81, 301–315.
Stewart, H. B., and Wendroff, B. (1984). Two-phase flow: models and methods.J. Comp. Phys. 56, 363–409.
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Keyfitz, B.L., Lopes-Filho, M.d.C. A geometric study of shocks in equations that change type. J Dyn Diff Equat 6, 351–393 (1994). https://doi.org/10.1007/BF02218855
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DOI: https://doi.org/10.1007/BF02218855