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Waves in Hyperbolic Systems

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Classical and Relativistic Rational Extended Thermodynamics of Gases

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Abstract

Wave propagation phenomena give us an important mean to check the validation of a nonequilibrium thermodynamics theory. In this chapter, we present a short review on the modern theory of wave propagation for hyperbolic systems. Firstly, we present the theory of linear waves emphasizing the role of the dispersion relation. The high frequency limit in the dispersion relation is also studied. Secondly, nonlinear acceleration waves are discussed together with the transport equation and the critical time. Thirdly, we present the main results concerning shock waves as a particular class of weak solutions and the admissibility criterion to select physical shocks (Lax condition, entropy growth condition, and Liu condition). Fourth, we discuss traveling waves, in particular, shock waves with structure. The subshock formation is particularly interesting. The Riemann problem and the problem of the large-time asymptotic behavior are also discussed. Lastly, we present toy models to show explicitly some interesting features obtained here.

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Notes

  1. 1.

    For simplicity, we use the symbols g and g 0 for the values of a generic quantity g evaluated at Γ with the condition that ϕ → 0− and ϕ → 0+, respectively.

  2. 2.

    As the balance laws satisfy the Galilean invariance, it is possible to adopt the frame moving with the shock velocity. For such an observer the wave appears stationary: φ = x.

  3. 3.

    We adopt different definition of the sign of the entropy from the one adopted in [242].

References

  1. I. Müller, T. Ruggeri, Extended Thermodynamics, 1st edn. (Springer, New York, 1993)

    Book  MATH  Google Scholar 

  2. G. Boillat, T. Ruggeri, Moment equations in the kinetic theory of gases and wave velocities. Continuum Mech. Thermodyn. 9, 205 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Boillat, T. Ruggeri, On the evolution law of the weak discontinuities for hyperbolic quasi-linear systems. Wave Motion 1, 149 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Ruggeri, Stability and discontinuity waves for symmetric hyperbolic systems, in Non-Linear Wave Motion ed. by A. Jeffrey (Longman Press, New York, 1989), pp.148–161

    Google Scholar 

  5. A. Muracchini, T. Ruggeri, L. Seccia, Dispersion relation in the high frequency limit and non-linear wave stability for hyperbolic dissipative systems. Wave Motion 15, 143 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Z. Banach, W. Larecki, T. Ruggeri, Dispersion relation in the limit of high frequency for a hyperbolic system with multiple eigenvalues. Wave Motion 51, 955 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Boillat, T. Ruggeri, On the shock structure problem for hyperbolic system of balance laws and convex entropy. Continuum Mech. Thermodyn. 10, 285 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Gilbarg, D. Paolucci, The structure of shock waves in the continuum theory of fluids. J. Rat. Mech. Anal. 2, 617 (1953)

    MathSciNet  MATH  Google Scholar 

  9. T. Ruggeri, S. Simić, On the hyperbolic system of a mixture of eulerian fluids: a comparison between single and multi-temperature models. Math. Meth. Appl. Sci. 30, 827 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Boillat, T. Ruggeri, Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rational Mech. Anal. 137, 305 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14, 249 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Lou, T. Ruggeri, Acceleration waves and weak Shizuta-Kawashima condition. Suppl. Rend. Circ. Mat. Palermo 78, 187 (2006)

    MathSciNet  MATH  Google Scholar 

  13. G. Boillat, in Recent Mathematical Methods in Nonlinear Wave Propagation. CIME Course, Lecture Notes in Mathematics, vol. 1640, ed. by T. Ruggeri (Springer, Berlin, 1995), pp. 103–152

    Google Scholar 

  14. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 325 (Springer, Berlin, 2010)

    Google Scholar 

  15. G. Boillat, La Propagation des Ondes (Gauthier-Villars, Paris, 1965)

    MATH  Google Scholar 

  16. G. Boillat, Chocs caractéeristiques, C. R. Acad Sci. Paris A 274, 1018 (1972)

    MathSciNet  MATH  Google Scholar 

  17. V.D. Sharma, Quasilinear Hyperbolic Systems, Compressible Flows, and Waves. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 142. (CRC Press, Boca Raton, 2010)

    Google Scholar 

  18. P.O.K. Krehl, History of Shock Waves, Explosions and Impact: A Chronological and Biographical Reference (Springer, Heidelberg, 2009)

    Google Scholar 

  19. D.G. Morris, An investigation of the shock-induced transformation of graphite to diamond. J. Appl. Phys. 51, 2059 (1980)

    Article  Google Scholar 

  20. J. Vink, Supernova remnants: the X-ray perspective. Astron Astrophys Rev. 20, 49 (2012)

    Article  Google Scholar 

  21. C.G. Chaussy, F. Eisenberger, B. Forssmann, Extracorporeal shockwave lithotripsy (ESWL): a chronology. J. Endourology 21, 1249 (2007)

    Article  Google Scholar 

  22. T. Ruggeri, S. Taniguchi, Shock waves in hyperbolic systems of non-equilibrium thermodynamics, in Applied Wave Mathematics II, ed. by A. Berezovski, T. Soomere (Springer, Cham, 2019), pp.167–186

    Chapter  Google Scholar 

  23. N. Zhao, A. Mentrelli, T. Ruggeri, M. Sugiyama, Admissible shock waves and shock-induced phase transitions in a van der Waals fluid. Phys. Fluids 23, 086101 (2011)

    Article  MATH  Google Scholar 

  24. P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. CBMS-NSF, Regional Conference Series in Applied Mathematics, vol. 11 (SIAM, New York, 1973)

    Google Scholar 

  25. T.-P. Liu, The entropy condition and the admissibility of shocks. J. Math. Anal. Appl. 53, 78 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  26. T.-P. Liu, Admissible solutions of hyperbolic conservation laws. Mem. Am. Math. Soc. 240, 12 (1981)

    MathSciNet  Google Scholar 

  27. T.-P. Liu, T. Ruggeri, Entropy production and admissibility of shocks. Acta Math. Appl. Sinica, English Series. 19, 1 (2003)

    Google Scholar 

  28. S. Taniguchi, A. Mentrelli, T. Ruggeri, M. Sugiyama, N. Zhao, Prediction and simulation of compressive shocks with lower perturbed density for increasing shock strength in real gases. Phys. Rev. E 82, 036324 (2010)

    Article  Google Scholar 

  29. N. Zhao, M. Sugiyama, T. Ruggeri, Phase transition induced by a shock wave in hard-sphere and hard-disc systems. J. Chem. Phys. 129, 054506 (2008)

    Article  Google Scholar 

  30. S. Taniguchi, A. Mentrelli, N. Zhao, T. Ruggeri, M. Sugiyama, Shock-induced phase transition in systems of hard spheres with internal degrees of freedom. Phys. Rev. E 81, 066307 (2010)

    Article  Google Scholar 

  31. Y. Zheng, N. Zhao, T. Ruggeri, M. Sugiyama, S. Taniguchi, Non-polytropic effect on shock-induced phase transitions in a hard-sphere system. Phys. Lett. A 374, 3315 (2010)

    Article  MATH  Google Scholar 

  32. S. Taniguchi, N. Zhao, M. Sugiyama, Shock-induced phase transitions from gas phase to solid phase. J. Phys. Soc. Japan 80, 083401 (2011)

    Article  Google Scholar 

  33. S. Taniguchi, M. Sugiyama, Shock-induced phase transitions in systems of hard spheres with attractive interactions. Acta Appl. Math. 122, 473 (2012)

    MATH  Google Scholar 

  34. G. Boillat, T. Ruggeri, Reflection and transmission of discontinuity waves through a shock wave, General theory including also the case of characteristic shocks. Proc. Roy. Soc. Edinb. 83A, 17 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  35. T. Ruggeri, Interaction between a discontinuity wave and a shock wave: critical time for the fastest transmitted wave, example of the polytropic fluid. Appl. Anal. 11, 103 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Pandey, V.D. Sharma, Interaction of a characteristic shock with a weak discontinuity in a non-ideal gas. Wave Motion 44, 346 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Mentrelli, T. Ruggeri, M. Sugiyama, N. Zhao, Interaction between a shock and an acceleration wave in a perfect gas for increasing shock strength. Wave Motion 45, 498 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. H. Alsmeyer, Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluid Mech. 74, 497 (1976)

    Article  Google Scholar 

  39. R. Becker, Stoßwelle und Detonation, Z. f. Physik 8, 321 (1922)

    Article  Google Scholar 

  40. H.W. Mott-Smith, The solution of the Boltzmann equation for a shock wave. Phys. Rev. 82, 885 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  41. K. Zoller, Zur Struktur des Verdichtungsstoßes. Zeitschr. f. Physik 130, 1 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  42. J.D. Foch, On higher order hydrodynamic theories of shock structure. Acta Physica Austriaca Suppl. 10, 123 (1973)

    Google Scholar 

  43. G.A. Bird, Aspects of the structure of strong shock waves. Phys. Fluids 13, 1172 (1970)

    Article  Google Scholar 

  44. T. Ruggeri, Breakdown of shock-wave-structure solutions. Phys. Rev. E 47, 4135 (1993)

    Article  MathSciNet  Google Scholar 

  45. W. Weiss, Continuous shock structure in extended thermodynamics. Phys. Rev. E 52, R5760 (1995)

    Article  Google Scholar 

  46. T. Ruggeri, Non existence of shock structure solutions for hyperbolic dissipative systems including characteristic shocks. Appl. Anal. 57, 23 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  47. D. Gilbarg, The existence and limit behavior of the one-dimensional shock layer. Am. J. Math. 73, 256 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  48. K. Zumbrun, W-A. Yong, Existence of relaxation shock profiles for hyperbolic conservation laws. SIAM J. Appl. Math. 60, 1565 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  49. S. Simić, Shock structure in continuum models of gas dynamics: stability and bifurcation analysis. Nonlinearity 22, 1337 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. S. Simić, Shock structure in the mixture of gases: stability and bifurcation of Equilibria, Chapter 8, in Thermodynamics—Kinetics of Dynamic Systems, ed. by J.C. Moreno Pirajãn (InTech, Rijeka, 2011)

    Google Scholar 

  51. P. Lax, Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  52. J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, Berlin, 1994)

    Book  MATH  Google Scholar 

  53. D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves (Cambridge University, Cambridge, 1999)

    Book  MATH  Google Scholar 

  54. E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer, Berlin, 2009)

    Book  MATH  Google Scholar 

  55. M. Dumbser, W. Boscheri, M. Semplice, G. Russo, Central weighted ENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes. SIAM J. Sci. Comput. 39, A2564 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  56. C. Klingenberg, Relaxation-projection schemes, the ultimate approximate Riemann solvers, in Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, ed. by G.V. Demidenko, E. Romenski, E. Toro, M. Dumbser (Springer, Cham, 2020), p. 237. ISBN 978-3-030-38869-0

    Google Scholar 

  57. T.-P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws. Commun. Pure Appl. Math. 30, 767 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  58. T.-P. Liu, Large-time behavior of solutions of initial and initial-boundary value problems of a general system of hyperbolic conservation laws. Commun. Math. Phys. 55, 163 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  59. A. Mentrelli, T. Ruggeri, Asymptotic behavior of Riemann and Riemann with Structure problems for a 2×2 hyperbolic dissipative system. Suppl. Rend. Circ. Mat. Palermo II 78, 201 (2006)

    MathSciNet  MATH  Google Scholar 

  60. F. Conforto, A. Mentrelli, T. Ruggeri, Shock structure and multiple sub-shocks in binary mixtures of Eulerian fluids. Ric. Mat. 66, 221 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  61. V. Artale, F. Conforto, G. Martalò, A. Ricciardello, Shock structure and multiple sub-shocks in Grad 10-moment binary mixtures of monatomic gases. Ric. Mat. 68, 485 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  62. M. Bisi, F. Conforto, G. Martalò, Sub-shock formation in Grad 10-moment equations for a binary gas mixture. Continuum Mech. Thermodyn. 28, 1295 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  63. S. Taniguchi, T. Ruggeri, On the sub-shock formation in extended thermodynamics. Int. J. Non-Linear Mech. 99, 69 (2018)

    Article  Google Scholar 

  64. F. Brini, T. Ruggeri, On the Riemann problem in extended thermodynamics, in Proceedings of the 10th International Conference on Hyperbolic Problems (HYP2004); Osaka, September 13–17, 2004, vol. I (Yokohama Publication, New York, 2006), p. 319

    Google Scholar 

  65. F. Brini, T. Ruggeri, On the Riemann problem with structure in extended thermodynamics, in Supplementary Rendiconti del Circolo Matematico di Palermo “Non Linear Hyperbolic Fields and Waves, A tribute to Guy Boillat”, Serie II, vol. 78 (2006), p. 21

    Google Scholar 

  66. T.-P. Liu, Nonlinear hyperbolic-dissipative partial differential equations, in Recent Mathematical Methods in Nonlinear Wave Propagation, Lecture Notes in Mathematics, vol. 1640, ed. by T. Ruggeri (Springer, Berlin, 1996), pp. 103–136

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Research Center on Applied Mathematics, University of Bologna, Bologna, Italy

    Tommaso Ruggeri

  2. Nagoya Institute of Technology, Nagoya, Japan

    Masaru Sugiyama

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  1. Tommaso Ruggeri
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  2. Masaru Sugiyama
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Ruggeri, T., Sugiyama, M. (2021). Waves in Hyperbolic Systems. In: Classical and Relativistic Rational Extended Thermodynamics of Gases. Springer, Cham. https://doi.org/10.1007/978-3-030-59144-1_3

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