Book Review
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MathSciNet review:
1567507
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Book Information:
Author:
Joel Smoller
Title:
Shock waves and reaction-diffusion equations
Additional book information:
A Series of Comprehensive Studies in Mathematics, Vol. 258, Springer-Verlag, New York, 1983, xx + 581 pp., $39.00. ISBN 0-3879-0752-1.
N. S. Bahvalov, The existence in the large of a regular solution of a quasilinear hyperbolic system, Ž. Vyčisl. Mat i Mat. Fiz. 10 (1970), 969–980 (Russian). MR 279443
Charles C. Conley and Joel A. Smoller, On the structure of magnetohydrodynamic shock waves, Comm. Pure Appl. Math. 27 (1974), 367–375. MR 368586, DOI 10.1002/cpa.3160270306
Edward Conway and Joel Smoller, Clobal solutions of the Cauchy problem for quasi-linear first-order equations in several space variables, Comm. Pure Appl. Math. 19 (1966), 95–105. MR 192161, DOI 10.1002/cpa.3160190107
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948. MR 0029615
Michael G. Crandall, The semigroup approach to first order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108–132. MR 316925, DOI 10.1007/BF02764657
C. M. Dafermos, Characteristics in hyperbolic conservation laws. A study of the structure and the asymptotic behaviour of solutions, Nonlinear analysis and mechanics: Heriot-Watt Symposium (Edinburgh, 1976), Vol. I, Res. Notes in Math., No. 17, Pitman, London, 1977, pp. 1–58. MR 0481581
C. M. Dafermos, Hyperbolic systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 25–70. MR 725517
8. C. M. Dafermos, Quasilinear hyperbolic systems that result from conservation laws, Nonlinear Waves (S. Leibovich and A. R. Seebass, eds.), Cornell Univ. Press, Ithaca, N. Y., 1974.
C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), no. 6, 1097–1119. MR 457947, DOI 10.1512/iumj.1977.26.26088
Ennio De Giorgi, Su una teoria generale della misura $(r-1)$-dimensionale in uno spazio ad $r$ dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), 191–213 (Italian). MR 62214, DOI 10.1007/BF02412838
Ronald J. DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws, Indiana Univ. Math. J. 24 (1974/75), no. 11, 1047–1071. MR 410110, DOI 10.1512/iumj.1975.24.24088
Ronald J. DiPerna, Singularities of solutions of nonlinear hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 60 (1975/76), no. 1, 75–100. MR 393867, DOI 10.1007/BF00281470
R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, DOI 10.1007/BF00251724
Ronald J. DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws, Indiana Univ. Math. J. 24 (1974/75), no. 11, 1047–1071. MR 410110, DOI 10.1512/iumj.1975.24.24088
Björn Engquist and Stanley Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981), no. 154, 321–351. MR 606500, DOI 10.1090/S0025-5718-1981-0606500-X
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 194770, DOI 10.1002/cpa.3160180408
James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970. MR 0265767
J. M. Greenberg, Decay theorems for stopping-shock problems, J. Math. Anal. Appl. 50 (1975), 314–324. MR 364882, DOI 10.1016/0022-247X(75)90025-6
20.J. M. Greenberg, The Cauchy problem for the quasilinear wave equations (unpublished preprint).
A. Harten, J. M. Hyman, and P. D. Lax, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR 413526, DOI 10.1002/cpa.3160290305
Eberhard Hopf, The partial differential equation $u_t+uu_x=\mu u_{xx}$, Comm. Pure Appl. Math. 3 (1950), 201–230. MR 47234, DOI 10.1002/cpa.3160030302
A. Jeffrey, Quasilinear hyperbolic systems and waves, Research Notes in Mathematics, No. 5, Pitman Publishing, London-San Francisco, Calif.-Melbourne, 1976. MR 0417585
Barbara Keyfitz Quinn, Solutions with shocks: An example of an $L_{1}$-contractive semigroup, Comm. Pure Appl. Math. 24 (1971), 125–132. MR 271545, DOI 10.1002/cpa.3160240203
25. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 127-243.
P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216
Peter D. Lax, The formation and decay of shock waves, Amer. Math. Monthly 79 (1972), 227–241. MR 298252, DOI 10.2307/2316618
Tai Ping Liu, Solutions in the large for the equations of nonisentropic gas dynamics, Indiana Univ. Math. J. 26 (1977), no. 1, 147–177. MR 435618, DOI 10.1512/iumj.1977.26.26011
Tai Ping Liu, Initial-boundary value problems for gas dynamics, Arch. Rational Mech. Anal. 64 (1977), no. 2, 137–168. MR 433017, DOI 10.1007/BF00280095
Tai Ping Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), no. 2, 135–148. MR 470508
32. T.-P. Liu, Admissible solutions to systems of conservation laws, Mem. Amer. Math. Soc. 240 (1982).
Andrew Majda and Stanley Osher, Numerical viscosity and the entropy condition, Comm. Pure Appl. Math. 32 (1979), no. 6, 797–838. MR 539160, DOI 10.1002/cpa.3160320605
Andrew Majda and James Ralston, Discrete shock profiles for systems of conservation laws, Comm. Pure Appl. Math. 32 (1979), no. 4, 445–482. MR 528630, DOI 10.1002/cpa.3160320402
A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR 748308, DOI 10.1007/978-1-4612-1116-7
François Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 489–507 (French). MR 506997
Takaaki Nishida, Global solution for an initial boundary value problem of a quasilinear hyperbolic system, Proc. Japan Acad. 44 (1968), 642–646. MR 236526
Takaaki Nishida and Joel A. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math. 26 (1973), 183–200. MR 330789, DOI 10.1002/cpa.3160260205
O. A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95–172. MR 0151737, DOI 10.1090/trans2/026/05
40. B. L. Rozhdestvensky and N. N. Yanenko, Quasilinear systems and their applications to the dynamics of gases, "Nauka", Moscow, 1968. (Russian)
Luc Tartar, The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 263–285. MR 725524
L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
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. MR 626623, DOI 10.1016/0022-0396(81)90055-3
Bram van Leer, An introduction to the article “Reminiscences about difference schemes” [J. Comput. Phys. 153 (1999), no. 1, 6–25; MR1703647 (2000h:65121)] by S. K. Godunov, J. Comput. Phys. 153 (1999), no. 1, 1–5. MR 1703646, DOI 10.1006/jcph.1999.6270
A. I. Vol′pert, Spaces $\textrm {BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338
Ronald J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), no. 1, 1–30. MR 719807
Peter D. Lax, The formation and decay of shock waves, Visiting scholars’ lectures (Texas Tech Univ., Lubbock, Tex., 1970/71), Math. Ser., No. 9, Texas Tech Press, Texas Tech Univ., Lubbock, Tex., 1971, pp. 107–139. MR 0367471
- 1.
- H. Bakhrarov, On the existence of regular solutions in the large for quasilinear hyperbolic systems, Zh. Vychisl. Mat. i Mat. Fiz. 10 (1970), 969-980. MR 0279443
- 2.
- C. Conley and J. Smoller, On the structure of magnetohydrodynamic shock waves, Comm. Pure Appl. Math. 28 (1974), 367-375. MR 368586
- 3.
- E. Conway and J. A. Smoller, Global solutions of the Cauchy problem for quasi-linear equations in several space variables, Comm. Pure Appl. Math. 19 (1966), 95-105. MR 192161
- 4.
- R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Interscience, New York, 1948. MR 29615
- 5.
- M. Crandall, The semigroup approach to first-order quasi-linear equations in several space variables, Israel J. Math. 12 (1972), 108-132. MR 316925
- 6.
- C. M. Dafermos, Characteristics in hyperbolic conservation laws. A study of the structure and the asymptotic behavior of solutions, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 1 (R. J. Knops, ed.), Pitman, London, 1977. MR 481581
- 7.
- C. M. Dafermos, Hyperbolic systems of conservation laws, Systems of Nonlinear Partial Differential Equations (J. M. Ball, ed.), Reidel, 1983. MR 725517
- 8.
- C. M. Dafermos, Quasilinear hyperbolic systems that result from conservation laws, Nonlinear Waves (S. Leibovich and A. R. Seebass, eds.), Cornell Univ. Press, Ithaca, N. Y., 1974.
- 9.
- C. M. Dafermos, Generalized characteristics and the structure of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), 1097-1119. MR 457947
- 10.
- E. DeGiorgi, Su una teoria generale della misura (r - 1)-dimensionale in uno spacio ad r dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), 191-213. MR 62214
- 11.
- R. J. DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws, Indiana Univ. Math. J. 24 (1975), 1047-1071. MR 410110
- 12.
- R. J. DiPerna, Singularities of solutions of nonlinear hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 60 (1975), 75-100. MR 393867
- 13.
- R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27-70. MR 684413
- 14.
- R. J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 24 (1975), 1047-1071. MR 410110
- 15.
- B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981), 321-351. MR 606500
- 16.
- H. Federer, Geometric measure theory, Springer, New York, 1969. MR 257325
- 17.
- J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. MR 194770
- 18.
- J. Glimm and P. D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Mem. Amer. Math. Soc. 101 (1970). MR 265767
- 19.
- J. M. Greenberg, Decay theorems for stopping-shock problems, J. Math. Anal. Appl. 50 (1975), 314-324. MR 364882
- 20.
- J. M. Greenberg, The Cauchy problem for the quasilinear wave equations (unpublished preprint).
- 21.
- A. Harten, J. Hyman and P. D. Lax, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), 297-322. MR 413526
- 22.
- E. Hopf, The partial differential equation u, Comm. Pure Appl. Math. 3 (1950), 201-230. MR 47234
- 23.
- A. Jeffrey, Quasilinear hyperbolic systems and waves, Pitman, London, 1976. MR 417585
- 24.
- B. Keyfitz, Solutions with shocks, an example of an L1-contractive semi-group, Comm. Pure Appl. 24 (1971), 125-132. MR 271545
- 25.
- N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 127-243.
- 26.
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537-566. MR 93653
- 27.
- P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS Regional Conf. Ser. Appl. Math., 11, SIAM, Philadelphia, 1973. MR 350216
- 28.
- P. D. Lax, The formation and decay of shock waves, Amer. Math. Monthly 79 (1972), 227-241. MR 298252
- 29.
- T.-P. Liu, Solutions in the large for the equations of non-isentropic gas dynamics, Indiana Univ. Math. J. 26 (1977), 147-177. MR 435618
- 30.
- T.-P. Liu, Initial-boundary value problems for gas dynamics, Arch. Rational Mech. Anal. 64 (1977), 137-168. MR 433017
- 31.
- T.-P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), 135-148. MR 470508
- 32.
- T.-P. Liu, Admissible solutions to systems of conservation laws, Mem. Amer. Math. Soc. 240 (1982).
- 33.
- A. Majda and S. Osher, Numerical viscosity and the entropy conditions, Comm. Pure Appl. Math. 32 (1979), 797-838. MR 539160
- 34.
- A. Majda and J. Ralston, Discrete shock profiles for systems of conservation laws, Comm. Pure Appl. Math. 32 (1979), 445-482. MR 528630
- 35.
- A. Majda, Compresible fluid flow and systems of conservation laws in several space variables, Preprint # 144, Center for Pure and Appl. Math., Univ. of California, Berkeley. MR 748308
- 36.
- F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Sci. Cl. 5 (1978), 69-102. MR 506997
- 37.
- T. Nishida, Global solutions for an initial-boundary value problem of a quasilinear hyperbolic system, Proc. Japan Acad. 44 (1968), 642-646. MR 236526
- 38.
- T. Nishida and J. A. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math. 26 (1973), 183-200. MR 330789
- 39.
- O. A. Oleinik, Discontinuous solutions of nonlinear differential equations, Uspekhi Mat. Nauk (N.S.) 12 (1957), no. 3 (75), 3-73; English transl. in Amer. Math. Soc. Transl. (2) 26 (1963), 95-172. MR 151737
- 40.
- B. L. Rozhdestvensky and N. N. Yanenko, Quasilinear systems and their applications to the dynamics of gases, "Nauka", Moscow, 1968. (Russian)
- 41.
- L. Tartar, The compensated compactness method applied to systems of conservation laws, Systems of Nonlinear Partial Differential Equations (J. M. Ball, ed.), Reidel, 1983. MR 725524
- 42.
- L. Tartar, Compensated compactness and applications to partial differential equations, Research Notes in Mathematics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 4 (R. J. Knops, ed.), Pitman, London, 1979. MR 584398
- 43.
- B. Temple, Solutions in the large for nonlinear hyperbolic conservation laws of gas dynamics, J. Differential Equations 41 (1981), 96-161. MR 626623
- 44.
- B. van Leer, Towards the ultimate conservative difference scheme. V: A second-order sequel to Godunov's method, J. Comput. Phys. 32 (1979), 101-136. MR 1703646
- 45.
- A. I. Vol'pert, The spaces BV and quasilinear equations, Math. USSR-Sb. 2 (1967), 257-267. MR 216338
- 46.
- R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. (to appear). MR 719807
- 47.
- P. D. Lax, Shock waves and entropy, Contributions to Nonlinear Functional Analysis (F. Zarantonello, ed.), Academic Press, 1971. MR 367471
Review Information:
Reviewer:
Ronald J. DiPerna
Journal:
Bull. Amer. Math. Soc.
11 (1984), 204-214
DOI:
https://doi.org/10.1090/S0273-0979-1984-15271-6