Abstract
We prove a result of existence and uniqueness of entropy weak solutions for two nonstrictly hyperbolic systems, both a nonconservative system of two equations
, and a conservative system of two equations
, where f: R → R is a given strictly convex function and \( a = \frac{d}{{du}}f \). We use the Volpert’s product ([19], see also Dal Maso — Le Floch — Murat [1]) and find entropy weak solutions u and w which have bounded variation while the solutions v are Borel measures. The equations for w and v can be viewed as linear hyperbolic equations with discontinuous coefficients.
This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.
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References
G. Dal Maso, Ph. Le Floch, F.Murat, Definition and weak stability of a nonconserva-tive product, Internal Report, Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau (FRANCE) (to appear).
G. Dal Maso, Ph Le Floch, F. Murat, Definition et stabilité faible d’un produit noncon-servatif, Note C.R. Acad. Sc. Paris (to appear).
R. DiPerna, Uniqueness of solutions of hyperbolic conservation laws, Ind. Univ. Math. J., 28 (1979), pp. 137–188.
R.J. DiPerna, P.L. Lions, Ordinary differential equation, transport theory and Sobolev spaces, (to appear).
R. Di Perna, A. Majda, The validity of nonlinear geometric optics for weak solutions of conservation laws, Comm. Math. Phys., 98 (1985), pp. 313–347.
J. Glimm, Solutions in the large for nonlinear systems of equations, Comm. Pure Appl. Math., 18 (1965), pp. 697–715.
B. Keyfitz, A viscosity approximation to a system of conservation laws with no classical Riemann solution, to appear in Proc. of Int. Conf. on Hyp. Problems, Bordeaux 1988.
D.J. Korchinski, Solution of a Riemann problem for a 2 × 2 system of conservation laws possessing no classical weak solution, Ph. D. Thesis Adelphi University (1977).
H. Kranzer, (to appear).
N. Kruskov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), pp. 127–243.
P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS monograph N 0 11, SIAM (1973).
Ph. Le Floch, Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form, Comm. Part. diff. Eq., 13 (6) (1988), pp. 669–727.
Ph. Le Floch, Solutions faibles entropiques des systèmes hyperboliques nonlinéaires sous forme nonconservative, Note CR. Acad. Sc. Paris, t. 306, Série 1 (1988), pp. 181–186.
Ph. Le Floch, Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form, in Proc. of the Second Int. Conf. on Nonlin. Hyp. Problems, (Aachen FRG) (1988), pp. 362–373.
Ph. Le Floch, Shock waves for nonlinear hyperbolic systems in nonconservative form, IMA series, University of Minnesota, USA, (to appear).
J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York (1983).
H.B. Stewart, B. Wendroff, Two-phase flow: models and methods, J. Comp. Phys., 56 (1984), pp. 363–409.
J. Trangenstein, P. Colella, A higher-order godunov method for modeling finite deformations in elastic-plastic solids.
A.I. Volpert, The space BV and quasilinear equations, Math. USSR Sb. 2 (1967), pp. 225–267.
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Le Floch, P. (1990). An Existence and Uniqueness Result for Two Nonstrictly Hyperbolic Systems. In: Keyfitz, B.L., Shearer, M. (eds) Nonlinear Evolution Equations That Change Type. The IMA Volumes in Mathematics and Its Applications, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9049-7_10
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DOI: https://doi.org/10.1007/978-1-4613-9049-7_10
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