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Systems of conservation laws with invariant submanifolds. (English) Zbl 0559.35046
The author considers a quasilinear first order partial differential system of conservation laws in one space dimension: \(\partial_ tU+\partial_ xF(U)=0\) where U and F are \(R^ N\) vectors; \(U\equiv U(x,t)\). The aim of the paper is to characterize necessary and sufficient conditions on the geometry of a wave curve in order that a shock wave curve coincides with its associate rarefaction wave curve. Particular emphasis is devoted to the case when \(U\in R^ 2\).
Reviewer: T.Ruggeri

MSC:
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
76S05 Flows in porous media; filtration; seepage
76T99 Multiphase and multicomponent flows
35L80 Degenerate hyperbolic equations
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[1] Rutherford Aris and Neal R. Amundson, Mathematical methods in chemical engineering, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Volume 2: First-order partial differential equations with applications; Prentice-Hall International Series in the Physical and Chemical Engineering Sciences.
[2] R. Courant and K. O. Friedricks, Supersonic flow and shock wuves, Wiley, New York, 1948.
[3] James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697 – 715. · Zbl 0141.28902 · doi:10.1002/cpa.3160180408 · doi.org
[4] E. Isaacson, Global solution of a Riemann problem for a non-strictly hyperbolic system of conservation laws arising in enhanced oil recovery, J. Comput. Phys. (to appear).
[5] F. Helfferich and G. Klein, Multicomponent chromatography, Marcel Dekker, New York, 1970.
[6] Barbara L. Keyfitz and Herbert C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1979/80), no. 3, 219 – 241. · Zbl 0434.73019 · doi:10.1007/BF00281590 · doi.org
[7] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537 – 566. · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 · doi.org
[8] Peter Lax, Shock waves and entropy, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 603 – 634. · Zbl 0268.35014
[9] T. P. Liu and C. H. Wang, On a hyperbolic system of conservation laws which is not strictly hyperbolic, MRC Technical Summary Report #2184, December 29, 1980.
[10] D. W. Peaceman, Fundamentals of numerical reservoir simulation, Elsevier North-Holland, New York. · Zbl 0204.28001
[11] H. Rhee, R. Aris and N. R. Amundson, On the theory of multicomponent chromatography. Philos. Trans. Roy. Soc. London Ser A 267 (1970), 419-455. · Zbl 0233.76157
[12] Blake Temple, Global solution of the Cauchy problem for a class of 2\times 2 nonstrictly hyperbolic conservation laws, Adv. in Appl. Math. 3 (1982), no. 3, 335 – 375. · Zbl 0508.76107 · doi:10.1016/S0196-8858(82)80010-9 · doi.org
[13] J. Blake Temple, Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics, J. Differential Equations 41 (1981), no. 1, 96 – 161. · Zbl 0476.76070 · doi:10.1016/0022-0396(81)90055-3 · doi.org
[14] K. N. Chueh, C. C. Conley, and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26 (1977), no. 2, 373 – 392. · Zbl 0368.35040 · doi:10.1512/iumj.1977.26.26029 · doi.org
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