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Nonlinear Hyperbolic Equations

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Book cover Partial Differential Equations III

Part of the book series: Applied Mathematical Sciences ((AMS,volume 117))

Abstract

Here we study nonlinear hyperbolic equations, with emphasis on quasi-linear systems arising from continuum mechanics, describing such physical phenomena as vibrating strings and membranes and the motion of a compressible fluid, such as air.

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References

  1. S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Birkhäuser, Boston, 1995.

    Book  MATH  Google Scholar 

  2. A. Anile, J. Hunter, P. Pantano, and G. Russo, Ray Methods for Nonlinear Waves in Fluids and Plasmas, Longman, New York, 1993.

    MATH  Google Scholar 

  3. S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly 87(1980), 359–370.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Ball (ed.), Systems of Nonlinear Partial Differential Equations, Reidel, Boston, 1983.

    MATH  Google Scholar 

  5. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3-d Euler equations, Comm. Math. Phys. 94(1984), 61–66.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Beals, Propagation and Interaction of Singularities in Nonlinear HyperbolicProblems, Birkhäuser, 1989.

    Book  MATH  Google Scholar 

  7. M. Beals and M. Bezard, Low regularity solutions for field equations, Preprint, 1995.

    Google Scholar 

  8. M. Beals, R. Melrose, and J. Rauch (eds.), Microlocal Analysis and Nonlinear Waves, IMA Vols, in Math, and its Appl., Vol. 30, Springer-Verlag, New York, 1991.

    MATH  Google Scholar 

  9. J. Bony, Calcul symbolique et propagation des singularities pour les équations aux dérivées nonlinéaires, Ann. Sci. Ecole Norm. Sup. 14(1981), 209–246.

    MathSciNet  MATH  Google Scholar 

  10. P. Brenner and W. von Wahl, Global classical solutions of nonlinear wave equations, Math. Zeit. 176(1981), 87–121.

    Article  MATH  Google Scholar 

  11. R. Bryant, S. Chern, R. Gardner, H. Goldschmidt, and P. Griffiths, Exterior Differential Systems, MSRI Publ. #18, Springer-Verlag, New York, 1991.

    Book  MATH  Google Scholar 

  12. R. Caflisch, A simplified version of the abstract Cauchy-Kowalevski theorem with weak singularity, Bull. AMS 23(1990), 495–500.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Carasso, M. Rascle, and D. Serre, Etude d’un modèle hyperbolique en dynamique des cables, Math. Mod. Numer. Anal. 19(1985), 573–599.

    MathSciNet  MATH  Google Scholar 

  14. C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994.

    MATH  Google Scholar 

  15. Y. Choquet-Bruhat, Theoreme d’existence pour certains systèmes d’équations aux derivées partielles non linéaires, Acta Math. 88(1952), 141–225.

    Article  MathSciNet  Google Scholar 

  16. A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, New York, 1979.

    Book  MATH  Google Scholar 

  17. D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, CPAM 39(1986), 267–282.

    MathSciNet  MATH  Google Scholar 

  18. K. Chueh, C. Conley, and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Math. J. 26(1977), 372–411.

    Article  MathSciNet  Google Scholar 

  19. C. Conley and J. Smoller, Shock waves as limits of progressive wave solutions of higher order equations, CPAM 24(1971), 459–472.

    MathSciNet  MATH  Google Scholar 

  20. C. Conley and J. Smoller, On the structure of magnetohydrodynamic shock waves, J. Math. Pures et Appl. 54(1975), 429–444.

    MathSciNet  Google Scholar 

  21. E. Conway and J. Smoller, Global solutions of the Cauchy problem for quasilinear first order equations in several space variables, CPAM 19(1966), 95–105.

    MathSciNet  MATH  Google Scholar 

  22. R. Courant and K. Friedrichs, Supersonic Flow and Shock Waves, Wiley, New York, 1948.

    MATH  Google Scholar 

  23. C. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Rat. Mech. Anal. 52(1973), 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  24. C. Dafermos, Hyperbolic systems of conservation laws, pp. 25–70 in [Ba].

    Google Scholar 

  25. C. Dafermos and R. DiPerna, The Riemann problem for certain classes of hyperbolic conservation laws, J. Diff. Eqs. 20(1976), 90–114.

    Article  MathSciNet  MATH  Google Scholar 

  26. C. Dafermos and W. Hrusa, Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics, Arch. Rat. Mech. Anal. 87(1985), 267–292.

    Article  MathSciNet  MATH  Google Scholar 

  27. P. Dionne, Sur les problèmes de Cauchy bien posés, J. Anal. Math. 10(1962–63), 1–90.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. DiPerna, Existence in the large for nonlinear hyperbolic conservation laws, Arch. Rat. Mech. Anal. 52(1973), 244–257.

    Article  MathSciNet  MATH  Google Scholar 

  29. R. DiPerna, Singularities of solutions of nonlinear hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal. 60(1975), 75–100.

    Article  MathSciNet  MATH  Google Scholar 

  30. R. DiPerna, Uniqueness of solutions of conservation laws, Indiana Math. J. 28(1979), 244–257.

    Article  MathSciNet  Google Scholar 

  31. R. DiPerna, Convergence of approximate solutions to conservation laws, Arch.Rat. Mech. Anal. 82(1983), 27–70.

    Article  MathSciNet  MATH  Google Scholar 

  32. R. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91(1983), 1–30.

    Article  MathSciNet  MATH  Google Scholar 

  33. R. DiPerna, Compensated compactness and general systems of conservation laws, Trans. AMS 292(1985), 383–420.

    Article  MathSciNet  Google Scholar 

  34. L. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Reg. Conf. Ser. #74, AMS, Providence, R. I., 1990.

    MATH  Google Scholar 

  35. J. Fehribach and M. Shearer, Approximately periodic solutions of the elastic string equations, Appl. Anal. 32(1989), 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  36. R. Foy, Steady state solutions of hyperbolic systems of conservation laws with viscosity terms, CPAM 17(1964), 177–188.

    MathSciNet  MATH  Google Scholar 

  37. H. Freistühler, Dynamical stability and vanishing viscosity: a case study of a

    Google Scholar 

  38. non-strictly hyperbolic system, CPAM 45(1992), 561–582.

    Google Scholar 

  39. A. Friedman, A new proof and generalizations of the Cauchy-Kowalevski theorem, Trans. AMS 98(1961), 1–20.

    Article  MATH  Google Scholar 

  40. K. Friedrichs and P. Lax, On symmetrizable differential operators, Proc. Symp.Pure Math. 10(1967) 128–137.

    MathSciNet  Google Scholar 

  41. K. Friedrichs and P. Lax, Systems of conservation laws with a convex extension, Proc. Natl. Acad. Sci. USA 68(1971), 1686–1688.

    Article  MathSciNet  MATH  Google Scholar 

  42. P. Garabedian, Partial Differential Equations, Wiley, New York, 1964.

    MATH  Google Scholar 

  43. P. Garabedian, Stability of Cauchy’s problem in space for analytic systems of arbitrary type, J. Math. Mech. 9(1960), 905–914.

    MathSciNet  MATH  Google Scholar 

  44. I. Gel’fand, Some problems in the theory of quasilinear equations, Usp. Mat.Nauk 14(1959), 87–115; AMS Transi. 29(1963), 295–381.

    Google Scholar 

  45. J. Glimm, Solutions in the large for nonlinear systems of equations, CPAM 18(1965), 697–715.

    MathSciNet  MATH  Google Scholar 

  46. J. Glimm, Nonlinear and stochastic phenomena: the grand challenge for partial differential equations, SIAM Review 33(1991), 626–643.

    Article  MathSciNet  MATH  Google Scholar 

  47. J. Glimm and P. Lax, Decay of Solutions of Systems of Nonlinear HyperbolicConservation Laws, Memoirs AMS #101, Providence, R. I., 1970.

    Google Scholar 

  48. M. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of Math. 132(1990), 485–509.

    Article  MathSciNet  MATH  Google Scholar 

  49. D. Hoff, Invariant regions for systems of conservation laws, TAMS 289(1985), 591–610.

    Article  MathSciNet  MATH  Google Scholar 

  50. D. Hoff, Global existence for ID compressible, isentropic Navier-Stokes equations with large initial data, TAMS 303(1987), 169–181.

    MathSciNet  MATH  Google Scholar 

  51. E. Hopf, The partial differential equation ut + uux — γuxx CPAM 3(1950), 201–230.

    MathSciNet  MATH  Google Scholar 

  52. L. Hörmander, Non-linear Hyperbolic Differential Equations. Lecture Notes, Lund Univ., 1986–87.

    Google Scholar 

  53. T. Hughes, T. Kato, and J. Marsden, Well-posed quasi-linear second order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rat. Mech. Anal. 63(1976), 273–294.

    MathSciNet  Google Scholar 

  54. T. Hughes and J. Marsden, A Short Course in Fluid Mechanics, Publish or Perish, Boston, 1976.

    MATH  Google Scholar 

  55. J. Joly, G. Metivier, and J. Rauch, Non linear oscillations beyond caustics, Pre-publication 94–14, IRMAR, Rennes, France, 1994.

    Google Scholar 

  56. D. Joseph, M. Renardy, and J. Saut, Hyperbolicity and change of type in the flow of viscoelastic fluids, Arch. Rat. Mech. Anal 87(1985), 213–251.

    Article  MathSciNet  MATH  Google Scholar 

  57. T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Springer LNM 448(1974), 25–70.

    Google Scholar 

  58. B. Keyfitz and H. Kranzer, Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two nonlinear conservation laws, J. Diff. Eqs. 27(1978), 444–476.

    Article  MathSciNet  MATH  Google Scholar 

  59. B. Keyfitz and H. Kranzer, A system of non-strictly hyperbolic conservation laws arising in elasticity theory, Arch. Rat. Mech. Anal. 72(1980), 219–241.

    Article  MathSciNet  MATH  Google Scholar 

  60. B. Keyfitz and H. Kranzer (eds.), Nonstrictly Hyperbolic Conservation Laws, Contemp. Math #60, AMS, Providence, R. I., 1987.

    MATH  Google Scholar 

  61. B. Keyfitz and M. Shearer (eds.), Nonlinear Evolution Equations that ChangeType, IMA Vol. in Math, and its Appl., Springer-Verlag, New York, 1990.

    MATH  Google Scholar 

  62. S. Kichenassamy, Nonlinear Wave Equations, Marcel Dekker, New York, 1995.

    Google Scholar 

  63. S. Klainerman, Global existence for nonlinear wave equations, CPAM 33(1980), 43–101.

    MathSciNet  MATH  Google Scholar 

  64. D. Kotlow, Quasilinear parabolic equations and first order quasilinear conservation laws with bad Cauchy data, /. Math. Anal. Appl. 35(1971), 563–576.

    Article  MathSciNet  MATH  Google Scholar 

  65. L. Landau and E. Lifshitz, Fluid Mechanics, Course ofTheoretical Physics, Vol. 6, Pergammon Press, New York, 1959.

    Google Scholar 

  66. P. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, CPAM 7(1954), 159–193.

    MathSciNet  MATH  Google Scholar 

  67. P. Lax, Hyperbolic systems of conservation laws II, CPAM 10(1957), 537–566.

    MathSciNet  MATH  Google Scholar 

  68. P. Lax, The Theory of Hyperbolic Equations, Stanford Lecture Notes, 1963.

    Google Scholar 

  69. P. Lax, Shock waves and entropy, pp. 603–634 in [Zar].

    Google Scholar 

  70. P. Lax, The formation and decay of shock waves, Amer. Math. Monthly 79(1972), 227–241.

    Article  MathSciNet  MATH  Google Scholar 

  71. P. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Reg. Conf. Ser. Appl. Math. #11, SIAM, 1973.

    Book  MATH  Google Scholar 

  72. W. Lindquist (ed.), Current Progress in Hyperbolic Systems: Riemann Problemsand Computations, Contemp. Math., Vol. 100, AMS, Providence, R. I., 1989.

    MATH  Google Scholar 

  73. T.-P. Liu, The Riemann problem for general 2×2 conservation laws, Trans. AMS 199(1974), 89–112.

    MATH  Google Scholar 

  74. T.-P. Liu, The Riemann problem for general systems of conservation laws, J. Diff.Eqs. 18(1975), 218–234.

    Article  MATH  Google Scholar 

  75. T.-P. Liu, Uniqueness of weak solutions of the Cauchy problem for general 2×2 conservation laws, J. Diff Eqs. 20(1976), 369–388.

    Article  MATH  Google Scholar 

  76. T.-P. Liu, Solutions in the large for the equations of non-isentropic gas dynamics, Indiana Math. J. 26(1977), 147–177.

    Article  MATH  Google Scholar 

  77. T.-P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57(1977), 135–148.

    Article  MathSciNet  MATH  Google Scholar 

  78. T.-P. Liu, Nonlinear Stability of Shock Waves for Viscous Conservation Laws, Memoirs AMS #328, Providence, R. I., 1985.

    Google Scholar 

  79. T.-P. Liu and J. Smoller, The vacuum state in isentropic gas dynamics, Adv. Appl.Math. 1(1980), 345–359.

    Article  MathSciNet  MATH  Google Scholar 

  80. A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in SeveralSpace Variables, Appl. Math. Sci. #53, Springer-Verlag, 1984.

    Book  MATH  Google Scholar 

  81. A. Majda, The Stability of Multi-dimensional Shock Fronts. Memoirs AMS, #275, Providence, R. I., 1983.

    Google Scholar 

  82. A. Majda, The Existence of Multi-dimensional Shock Fronts. Memoirs AMS, #281, Providence, R. I., 1983.

    Google Scholar 

  83. A. Majda, Mathematical fluid dynamics: the interaction of nonlinear analysis and modern applied mathematics, Proc. AMS Centennial Symp. (1988), 351–394.

    Google Scholar 

  84. A. Majda, The interaction of nonlinear analysis and modern applied mathematics, Proc. International Congress Math. Kyoto, Springer-Verlag, New York, 1991.

    Google Scholar 

  85. A. Majda and S. Osher, Numerical viscosity and the entropy condition, CPAM 32(1979), 797–838.

    MathSciNet  MATH  Google Scholar 

  86. A. Majda and R. Rosales, A theory for spontaneous Mach stem formation in reacting shock fronts. I, SIAM J. Appl. Math. 43(1983), 1310–1334;

    Article  MathSciNet  MATH  Google Scholar 

  87. A. Majda and R. Rosales, A theory for spontaneous Mach stem formation in reacting shock fronts. II, Studies inAppl. Math. 71(1984), 117–148.

    MathSciNet  MATH  Google Scholar 

  88. A. Majda and E. Thomann, Multi-dimensional shock fronts for second order waveequations, Comm. PDE 12(1988), 777–828.

    Article  MathSciNet  Google Scholar 

  89. R. Menikoff, Analogies between Riemann problems for 1-D fluid dynamics and 2-D steady supersonic flow, pp.225–240 in [Lind].

    Google Scholar 

  90. G. Metivier, Interaction de deux chocs pour un système de deux lois de conservation en dimension deux d’espace, TAMS 296(1986), 431–479.

    Article  MathSciNet  MATH  Google Scholar 

  91. G. Metivier, Stability of multi-dimensional weak shocks, Comm. PDE 15(1990), 983–1028.

    Article  MathSciNet  MATH  Google Scholar 

  92. C. Morawetz, An alternative proof of DiPerna’s theorem, CPAM 45(1991), 1081–1090.

    MathSciNet  Google Scholar 

  93. L. Nirenberg, An abstract form for the nonlinear Cauchy-Kowalevski theorem, J.Diff. Geom. 6(1972), 561–576.

    MathSciNet  MATH  Google Scholar 

  94. T. Nishida, Global solutions for an initial boundary value problem of a quasilinear hyperbolic system, Proc. Japan Acad. 44(1968), 642–646.

    Article  MathSciNet  MATH  Google Scholar 

  95. T. Nishida and J. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws, CPAM 26(1973), 183–200.

    MathSciNet  MATH  Google Scholar 

  96. H. Ockendon and A. Tayler, Inviscid Fluid Flows, Appl. Math. Sci. #43, Springer-Verlag, New York, 1983.

    MATH  Google Scholar 

  97. O. Oleinik, Discontinuous solutions of non-linear differential equations, UspekhiMat. Nauk. 12(1957), 3–73.

    MathSciNet  Google Scholar 

  98. O. Oleinik, Discontinuous solutions of non-linear differential equations, AMS Transi. 26, 95–172.

    Google Scholar 

  99. O. Oleinik, On the uniqueness of the generalized solution of the Cauchy problem for a nonlinear system of equations occurring in mechanics, Uspekhi Mat. Nauk. 12(1957), 169–176.

    MathSciNet  Google Scholar 

  100. L. Ovsjannikov, A nonlinear Cauchy problem in a scale of Banach spaces, SovietMath.Dokl. 12(1971), 1497–1502.

    Google Scholar 

  101. R. Pego and D. Serre, Instabilities in Glimm’s scheme for two systems of mixed type, SIAMJ. Numer. Anal 25(1988), 965–989.

    Article  MathSciNet  MATH  Google Scholar 

  102. J. Rauch, The u 5-Klein-Gordon equation, Pitman Res. Notes in Math. #53, pp. 335–364.

    Google Scholar 

  103. J. Rauch and M. Reed, Propagation of singularities for semilinear hyperbolic equations in one space variable, Ann. Math. 111(1980), 531–552.

    Article  MathSciNet  MATH  Google Scholar 

  104. M. Reed, Abstract Non-Linear Wave Equations, LNM #507, Springer-Verlag, New York, 1976.

    MATH  Google Scholar 

  105. P. Resibois and M. DeLeener, Classical Kinetic Theory of Fluids, Wiley, New York, 1977.

    Google Scholar 

  106. B. Rubino, On the vanishing viscosity approximation to the Cauchy problem for a 2 × 2 system of conservation laws, Ann. Inst. H. Poincaré (Analyse non linéaire) 10(1993), 627–656.

    MathSciNet  MATH  Google Scholar 

  107. D. Schaeffer and M. Shearer, The classification of 2 x 2 systems of non-strictly hyperbolic conservation laws with application to oil recovery, CPAM 40(1987), 141–178.

    MathSciNet  MATH  Google Scholar 

  108. D. Schaeffer and M. Shearer, Riemann problems for nonstrictly hyperbolic 2×2 systems of conservation laws, TAMS 304(1987), 267–306.

    MathSciNet  MATH  Google Scholar 

  109. I. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France 91(1963), 129–135.

    MathSciNet  MATH  Google Scholar 

  110. D. Serre, La compacité par compensation pour les systèmes hyperboliques non-linéaires de deux equations a une dimension d’espace, J. Math. Pures et Appl. 65(1986), 423–468.

    MATH  Google Scholar 

  111. J. Shatah, Weak solutions and development of singularities of the SU(2) σ -model, CPAM 49(1988), 459–469.

    MathSciNet  Google Scholar 

  112. M. Shearer, The Riemann problem for a class of conservation laws of mixed type, J. Diff. Eqs. 46(1982), 426–443.

    Article  MathSciNet  MATH  Google Scholar 

  113. M. Shearer, Elementary wave solutions of the equations describing the motion of an elastic string, SIAMJ. Math. Anal. 16(1985), 447–459.

    Article  MathSciNet  MATH  Google Scholar 

  114. M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rat. Math. Anal. 81(1983), 301–315.

    MathSciNet  MATH  Google Scholar 

  115. R. Smith, The Riemann problem in gas dynamics, TAMS 249(1979), 1–50.

    Article  MATH  Google Scholar 

  116. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.

    Book  MATH  Google Scholar 

  117. J. Smoller and J. Johnson, Global solutions for an extended class of hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal. 32(1969), 169–189.

    Article  MathSciNet  MATH  Google Scholar 

  118. W. Strauss, Nonlinear Wave Equations, CBMS Reg. Conf. Ser. #73, AMS, Providence, R. I., 1989.

    MATH  Google Scholar 

  119. M. Struwe, Semilinear wave equations, Bull. AMS 26(1992), 53–85.

    Article  MathSciNet  MATH  Google Scholar 

  120. L. Tartar, Compensated compactness and applications to PDE, pp. 136–212 in Research Notes in Mathematics, Nonlinear Analysis, and Mechanics, Heriot-Watt Symp. Vol. 4, ed. R. Knops, Pitman, Boston, 1979.

    Google Scholar 

  121. L. Tartar, The compensated compactness method applied to systems of conservation laws, pp. 263–285 in [Ba].

    Google Scholar 

  122. M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.

    Book  MATH  Google Scholar 

  123. B. Temple, Global solutions of the Cauchy problem for a class of 2 x 2 nonstrictly hyperbolic conservation laws, Adv. Appl. Math. 3(1982), 335–375.

    Article  MathSciNet  MATH  Google Scholar 

  124. A. Volpert, The spaces BV and quasilinear equations, Math. USSR Sb. 2(1967), 257–267.

    Google Scholar 

  125. B. Wendroff, The Riemann problem for materials with non-convex equations of state, I: Isentropic flow, J. Math. Anal. Appl. 38(1972), 454–466.

    Article  MathSciNet  MATH  Google Scholar 

  126. H. Weyl, Shock waves in arbitrary fluids, CPAM 2(1949), 103–122.

    MathSciNet  MATH  Google Scholar 

  127. E. Zarantoneilo (ed.), Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971.

    Google Scholar 

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  1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA

    Michael E. Taylor

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Taylor, M.E. (1996). Nonlinear Hyperbolic Equations. In: Partial Differential Equations III. Applied Mathematical Sciences, vol 117. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4190-2_4

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