Book Review
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MathSciNet review:
2985957
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Book Information:
Author:
Vishnu D. Sharma
Title:
Quasilinear hyperbolic systems, compressible flows, and waves
Additional book information:
Chapman \& Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 142. CRC Press,
Boca Raton, FL,
2010,
xiv + 268 pp.,
ISBN 978-1-4398-3690-3
Stuart S. Antman, Nonlinear problems of elasticity, 2nd ed., Applied Mathematical Sciences, vol. 107, Springer, New York, 2005. MR 2132247
Sylvie Benzoni-Gavage and Denis Serre, Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. First-order systems and applications. MR 2284507
Alberto Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova 110 (2003), 103–117. MR 2033003
R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Applied Mathematical Sciences, Vol. 21, Springer-Verlag, New York-Heidelberg, 1976. Reprinting of the 1948 original. MR 0421279
Demetrios Christodoulou, The Euler equations of compressible fluid flow, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 4, 581–602. MR 2338367, DOI 10.1090/S0273-0979-07-01181-0
Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377, DOI 10.1007/978-3-642-04048-1
Camillo De Lellis, Blowup of the BV norm in the multidimensional Keyfitz and Kranzer system, Duke Math. J. 127 (2005), no. 2, 313–339. MR 2130415, DOI 10.1215/S0012-7094-04-12724-1
Xia Xi Ding, Gui Qiang Chen, and Pei Zhu Luo, Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Comm. Math. Phys. 121 (1989), no. 1, 63–84. MR 985615
Ronald J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), no. 1, 1–30. MR 719807
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
S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
Peter D. Lax, Hyperbolic partial differential equations, Courant Lecture Notes in Mathematics, vol. 14, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2006. With an appendix by Cathleen S. Morawetz. MR 2273657, DOI 10.1090/cln/014
Pierre-Louis Lions, Benoît Perthame, and Panagiotis E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49 (1996), no. 6, 599–638. MR 1383202, DOI 10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5
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
Ralph Menikoff and Bradley J. Plohr, The Riemann problem for fluid flow of real materials, Rev. Modern Phys. 61 (1989), no. 1, 75–130. MR 977944, DOI 10.1103/RevModPhys.61.75
François Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 489–507 (French). MR 506997
Benoît Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 21, Oxford University Press, Oxford, 2002. MR 2064166
Jeffrey Rauch, BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one, Comm. Math. Phys. 106 (1986), no. 3, 481–484. MR 859822
B. L. Roždestvenskiĭ and N. N. Janenko, Systems of quasilinear equations and their applications to gas dynamics, Translations of Mathematical Monographs, vol. 55, American Mathematical Society, Providence, RI, 1983. Translated from the second Russian edition by J. R. Schulenberger. MR 694243, DOI 10.1090/mmono/055
Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779, DOI 10.1007/978-1-4612-0873-0
Eitan Tadmor, Jian-Guo Liu, and Athanasios Tzavaras (eds.), Hyperbolic problems: theory, numerics and applications, Proceedings of Symposia in Applied Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2009. Plenary and invited talks. MR 2640499, DOI 10.1090/psapm/067.1
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
Blake Temple and Robin Young, A paradigm for time-periodic sound wave propagation in the compressible Euler equations, Methods Appl. Anal. 16 (2009), no. 3, 341–363. MR 2650801, DOI 10.4310/MAA.2009.v16.n3.a5
G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Reprint of the 1974 original; A Wiley-Interscience Publication. MR 1699025, DOI 10.1002/9781118032954
References
- Stuart S. Antman, Nonlinear problems of elasticity, 2nd ed., Applied Mathematical Sciences, vol. 107, Springer, New York, 2005. MR 2132247 (2006e:74001)
- Sylvie Benzoni-Gavage and Denis Serre, Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007. First-order systems and applications. MR 2284507 (2008k:35002)
- Alberto Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova 110 (2003), 103–117. MR 2033003 (2005a:35186)
- R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Springer-Verlag, New York, 1976. Reprinting of the 1948 original; Applied Mathematical Sciences, Vol. 21. MR 0421279 (54 \#9284)
- Demetrios Christodoulou, The Euler equations of compressible fluid flow, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 4, 581–602 (electronic). MR 2338367 (2009g:76126), DOI 10.1090/S0273-0979-07-01181-0
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377, DOI 10.1007/978-3-642-04048-1
- Camillo De Lellis, Blowup of the BV norm in the multidimensional Keyfitz and Kranzer system, Duke Math. J. 127 (2005), no. 2, 313–339. MR 2130415 (2006j:35159), DOI 10.1215/S0012-7094-04-12724-1
- Xia Xi Ding, Gui Qiang Chen, and Pei Zhu Luo, Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Comm. Math. Phys. 121 (1989), no. 1, 63–84. MR 985615 (90d:65168)
- Ronald J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), no. 1, 1–30. MR 719807 (85i:35118)
- James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 0194770 (33:2976)
- S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257 (42:2159)
- Peter D. Lax, Hyperbolic partial differential equations, Courant Lecture Notes in Mathematics, vol. 14, New York University Courant Institute of Mathematical Sciences, New York, 2006. With an appendix by Cathleen S. Morawetz. MR 2273657 (2007h:35002)
- Pierre-Louis Lions, Benoît Perthame, and Panagiotis E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49 (1996), no. 6, 599–638. MR 1383202 (97e:35107)
- 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 (85e:35077)
- Ralph Menikoff and Bradley J. Plohr, The Riemann problem for fluid flow of real materials, Rev. Modern Phys. 61 (1989), no. 1, 75–130. MR 977944 (90a:35142), DOI 10.1103/RevModPhys.61.75
- François Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 489–507 (French). MR 506997 (80h:46043a)
- Benoît Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 21, Oxford University Press, Oxford, 2002. MR 2064166 (2005d:35005)
- Jeffrey Rauch, BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one, Comm. Math. Phys. 106 (1986), no. 3, 481–484. MR 859822 (87m:35139)
- B. L. Roždestvenskiĭ and N. N. Janenko, Systems of quasilinear equations and their applications to gas dynamics, Translations of Mathematical Monographs, vol. 55, American Mathematical Society, Providence, RI, 1983. Translated from the second Russian edition by J. R. Schulenberger. MR 694243 (85f:35127)
- Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779 (95g:35002)
- Eitan Tadmor, Jian-Guo Liu, and Athanasios Tzavaras (eds.), Hyperbolic problems: theory, numerics and applications, Proceedings of Symposia in Applied Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2009. Plenary and invited talks. MR 2640499 (2010k:35006)
- 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., 1979, pp. 136–212. MR 584398 (81m:35014)
- Blake Temple and Robin Young, A paradigm for time-periodic sound wave propagation in the compressible Euler equations, Methods Appl. Anal. 16 (2009), no. 3, 341–363. MR 2650801
- G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1999. Reprint of the 1974 original; A Wiley-Interscience Publication. MR 1699025 (2000c:35001)
Review Information:
Reviewer:
Helge Kristian Jenssen
Affiliation:
Penn State University
Email:
hkj1@psu.edu
Journal:
Bull. Amer. Math. Soc.
49 (2012), 591-596
DOI:
https://doi.org/10.1090/S0273-0979-2011-01356-8
Published electronically:
September 13, 2011
Review copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
