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Quasilinear hyperbolic systems. (English) Zbl 0435.35054

35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
35L67 Shocks and singularities for hyperbolic equations
35L60 First-order nonlinear hyperbolic equations
Full Text: DOI
[1] Antman, S., Liu, T.-P.: Traveling waves in hyperelastic rods. Quart. Appl. Math. (to appear) · Zbl 0408.73043
[2] Chester, W.: The quasi-cylindrical shock tube. Philos. Mag. (7)45, 1293-1304 (1954) · Zbl 0057.18601
[3] Chisnell, P.: The normal motion of a shock wave through a nonuniform one-dimensional medium. Proc. R. Soc. Edinburgh Sect. A323, 350-370 (1955) · Zbl 0068.19201 · doi:10.1098/rspa.1955.0223
[4] Conley, C.C., Smoller, J.A.: Shock waves as limits of progressive wave solutions of higher order equation. Comm. Pure Appl. Math.24, 459-472 (1971) · Zbl 0233.35063 · doi:10.1002/cpa.3160240402
[5] Courant, R., Friedrichs, K.O.: Supersonic flow and shock waves. New York: Interscience (1948) · Zbl 0041.11302
[6] Courant, R., Hilbert, D.: Methods of mathematical physics. Vol. II, Chapt. V. 6, pp. 464-471. Interscience Publishers (1962) · Zbl 0099.29504
[7] Dafermos, C.M.: The entropy rate admissibility criterion for solutions of hyperbolic conservation laws. J. Diff. Eq.14, 202-212 (1973) · Zbl 0262.35038 · doi:10.1016/0022-0396(73)90043-0
[8] DiPerna, R.: Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws. Indiana Univ. Math. J.24, 1047-1071 (1975) · Zbl 0309.35050 · doi:10.1512/iumj.1975.24.24088
[9] Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math.18, 697-715 (1965) · Zbl 0141.28902 · doi:10.1002/cpa.3160180408
[10] Glimm, J., Lax, P.D.: Decay of solutions of nonlinear hyperbolic conservation laws. Am. Math. Soc.101, (1970) · Zbl 0204.11304
[11] John, F.: Formation of singularities in one-dimensional nonlinear wave propagation. Comm. Pure Appl. Math.27, 377-405 (1974) · Zbl 0302.35064 · doi:10.1002/cpa.3160270307
[12] Lax, P.D.: Hyperbolic systems of conservation laws. II. Comm. Pure Appl. Math.10, 537-566 (1957) · Zbl 0081.08803 · doi:10.1002/cpa.3160100406
[13] Lax, P.D.: Shock waves and entropy. In: Contributions to nonlinear functional analysis. Zarantonello, E.H. (ed.), pp. 603-634. New York: Academic Press 1971
[14] Lax, P.D.: Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Math. Phys.5, 611-613 (1964) · Zbl 0135.15101 · doi:10.1063/1.1704154
[15] Liu, T.-P.: The Riemann problem for general systems of conservation laws. J. Diff. Eq.18, 218-234 (1975) · Zbl 0297.76057 · doi:10.1016/0022-0396(75)90091-1
[16] Liu, T.-P.: Solutions in the large for equations of non-isotropic gas dynamics. Indiana Univ. J.26, 147-177 (1977) · Zbl 0361.35056 · doi:10.1512/iumj.1977.26.26011
[17] Liu, T.-P.: Decay toN-waves of solutions of general systems of nonlinear hyperbolic conservation laws. Comm. Pure Appl. Math.30, 585-610 (1977) · Zbl 0357.35059 · doi:10.1002/cpa.3160300505
[18] Liu, T.-P.: Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws. Comm. Pure Appl. Math.30, 767-796 (1977) · Zbl 0358.35014 · doi:10.1002/cpa.3160300605
[19] Liu, T.-P.: The deterministic version of the Glimm scheme. Comm. Math. Phys.57, 135-148 (1977) · Zbl 0376.35042 · doi:10.1007/BF01625772
[20] Liu, T.-P.: The development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations. J. Diff. Eq. (to appear)
[21] Nishida, T.: Global solutions for an initial boundary value problem of a quasilinear hyperbolic system. Proc. Jpn. Acad.44, 642-646 (1968) · Zbl 0167.10301 · doi:10.3792/pja/1195521083
[22] Nishida, T., Smoller, J.A.: Solutions in the large for some nonlinear hyperbolic conservation laws. Comm. Pure Appl. Math.26, 183-200 (1973) · Zbl 0267.35058 · doi:10.1002/cpa.3160260205
[23] Riemann, B.: Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungswite. Göttinger Abhandlungen, Vol. 8, p. 43. Werke, Zte Aufl. Leipzig 157 (1892) · JFM 24.0021.04
[24] Wendroff, B.: Shock propagation in variable area ducts with phase changes: an extention of Chisnell’s method. J. Eng. Math.11, 273-286 (1977) · Zbl 0365.76072 · doi:10.1007/BF01535971
[25] Whitham, B.: Linear and nonlinear waves. New York: John Wiley 1974 · Zbl 0373.76001
[26] Concus, P., Proskurowski, W.: Numerical solution of a nonlinear hyperbolic equation by the random choice method (to appear in J. Comp. Phys.) · Zbl 0399.65068
[27] Hoffman, A.L.: A single fluid model for shock formation in MHD shock tubes. J. Plasma Phys.1, 192-207 (1967) · doi:10.1017/S0022377800003214
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