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Compensated Compactness

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Hyperbolic Conservation Laws in Continuum Physics

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 325))

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Abstract

Approximate solutions to hyperbolic systems of conservation laws may be generated in a variety of ways: by the method of vanishing viscosity, through difference approximations, by relaxation schemes, etc. The topic for discussion in this chapter is whether solutions may be constructed as limits of sequences of approximate solutions that are only bounded in some L p space. Since the systems are nonlinear, the difficulty lies in that the construction schemes are generally consistent only when the sequence of approximating solutions converges strongly, whereas the assumed L p bounds only guarantee weak convergence: Approximate solutions may develop high frequency oscillations of finite amplitude which play havoc with consistency. The aim is to demonstrate that entropy inequalities may save the day by quenching rapid oscillations thus enforcing strong convergence of the approximating solutions. Some indication of this effect was alluded in Section 1.9.

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Notes

  1. Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Sc1. Fis. Mat. 5 (1978), 489–507.

    MathSciNet  MATH  Google Scholar 

  2. Tartar, L.C.: Cours Peccot, Collège de France 1977.

    Google Scholar 

  3. Tartar, L.C.: Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol IV, pp. 136–212, ed. R.J. Knops. London: Pitman, 1979.

    Google Scholar 

  4. Tartar, L.C.: The compensated compactness method applied to systems of conservation laws. Systems of Nonlinear Partial Differential Equations, pp. 263–285, ed. J.M. Ball. Dordrecht: D. Reidel, 1983.

    Google Scholar 

  5. DiPerna, R.J.: Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27–70.

    MathSciNet  MATH  Google Scholar 

  6. Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS Regional Conference Series in Mathematics No. 74. Providence: American Mathematical Society, 1990.

    Google Scholar 

  7. Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations. Paris: Springer, 1997.

    MATH  Google Scholar 

  8. Mâlek, J., Necas, J. and M. Rokyta: Weak and Measure-Valued Solutions to Evolutionary PDEs. London: Chapman und Hall, 1996.

    Google Scholar 

  9. Taylor, M.E.: Partial Differential Equations II1. New York: Springer, 1996.

    Google Scholar 

  10. Young, L.C.: Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, Classe III, 30 (1937), 212–234.

    MATH  Google Scholar 

  11. Ball, J.M.: A version of the fundamental theorem for Young measures. Partial Differential Equations and Continuum Models of Phase Transitions, pp. 241–259, ed. M. Rascle, D. Serre and M. Slemrod. Lecture Notes in Physics No. 344. Berlin: Springer, 1989.

    Google Scholar 

  12. Murat, F.: L’ injection du cône positif de H-1 dans W-i,y est compacte pour tout q 2. J. Math. Pures Appl. 60 (1981), 309–322.

    MathSciNet  MATH  Google Scholar 

  13. DiPerna, R.J.: Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985), 223–270.

    MathSciNet  MATH  Google Scholar 

  14. Chen, G.-Q. and H. Frid: Asymptotic Stability and Decay of Solutions of Conservation Laws. Lecture Notes, Northwestern U., 1996.

    Google Scholar 

  15. Chen, G.-Q. and P.G. Lefloch: Compressible Euler equations with general pressure law and related equations. Arch. Rational Mech. Anal. (To appear).

    Google Scholar 

  16. Demengel, F. and D. Serre: Nonvanishing singular parts of measure-valued solutions for scalar hyperbolic equations. Comm. PDE 16 (1991), 221–254.

    Google Scholar 

  17. Frid, H.: Initial-boundary value problems for conservation laws. J. Diff. Eqs. 128 (1996), 1–45.

    Article  MathSciNet  MATH  Google Scholar 

  18. Poupaud, F., Rascle, M. and J.P. Vila: Global solutions to the isothermal Euler-Poisson system with arbitrarily large data. J. Diff. Eqs. 123 (1995), 93–121.

    Article  MathSciNet  MATH  Google Scholar 

  19. Slemrod, M.: Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 81 (1983), 301–315.

    MathSciNet  MATH  Google Scholar 

  20. Slemrod, M.: Dynamic phase transitions in a van der Waals fluid. J. Diff. Eqs. 52 (1984), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  21. Szepessy, A.: Measure-valued solutions of scalar conservation laws with boundary conditions. Arch. Rational Mech. Anal. 107 (1989), 181–193.

    MathSciNet  MATH  Google Scholar 

  22. Szepessy, A.: An existence result for scalar conservation laws using measure valued solutions. Comm. PDE 14 (1989), 1329–1350.

    Article  MathSciNet  MATH  Google Scholar 

  23. Tartar, L.C.: Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol IV, pp. 136–212, ed. R.J. Knops. London: Pitman, 1979.

    Google Scholar 

  24. Vecchi, I.: A note on entropy compactification for scalar conservation laws. Nonlinear Analysis 15 (1990), 693–6695.

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, T.-P.: Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108 (1987), 153175.

    Google Scholar 

  26. Whitham, G.B.: Linear and Nonlinear Waves. New York: Wiley-Interscience, 1974.

    MATH  Google Scholar 

  27. Chen, G.-Q. and T.-P. Liu: Zero relaxation and dissipation limits for hyperbolic conservation laws. Comm. Pure Appl. Math. 46 (1993), 755–781.

    MATH  Google Scholar 

  28. Jin, S. and Z. Xin: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48 (1995), 235–276.

    MathSciNet  MATH  Google Scholar 

  29. Chen, G.-Q. and T.-P. Liu: Zero relaxation and dissipation limits for hyperbolic conservation laws. Comm. Pure Appl. Math. 46 (1993), 755–781.

    Google Scholar 

  30. Collet, J.F. and M. Rascle: Convergence of the relaxation approximation to a scalar nonlinear hyperbolic equation arising in chromatography. ZAMP 47 (1996), 400–409.

    Article  MathSciNet  MATH  Google Scholar 

  31. Coquel, F. and B. Perthame: Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Num. Anal. 35 (1998), 2223–2249.

    Article  MathSciNet  MATH  Google Scholar 

  32. Klingenberg, C. and Y. Lu: Cauchy problem for hyperbolic conservation laws with a relaxation tern1. Proc. Royal Soc. Edinburgh, 126A (1996), 821–828.

    Article  MathSciNet  Google Scholar 

  33. Lattanzio, C. and P. Marcati: The zero relaxation limit for the hydrodynamic Whitham traffic flow model. J. Diff. Eqs. 141 (1997), 150–178.

    Article  MathSciNet  MATH  Google Scholar 

  34. Lu, Y.-G. and C. Klingenberg: The Cauchy problem for hyperbolic conservation laws with three equations. J. Math. Anal. Appl. 202 (1996), 206–216.

    Article  MathSciNet  MATH  Google Scholar 

  35. Marcati, P. and R. Natalini: Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation. Arch. Rational Mech. Anal. 129 (1995), 129–145.

    MathSciNet  MATH  Google Scholar 

  36. Marcati, P. and R. Natalini: Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation. Arch. Rational Mech. Anal. 129 (1995), 129–145.

    MathSciNet  MATH  Google Scholar 

  37. Natalini, R.: Convergence to equilibrium for the relaxation approximations of conservation laws. Comm. Pure Appl. Math. 49 (1996), 795–823.

    MathSciNet  MATH  Google Scholar 

  38. Tveito A. and R. Winther: On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term. SIAM J. Math. Anal. 28 (1997), 136–161.

    MathSciNet  MATH  Google Scholar 

  39. Yong, W.-A.: Boundary conditions for hyperbolic systems with stiff source terms. Arch. Rational Mech. Anal. (To appear).

    Google Scholar 

  40. Natalini, R.: Recent results on hyperbolic relaxation problems. Analysis of Systems of Conservation Laws, pp. 128–198, ed. H. Freistühler. London: Chapman and Hall/CRC, 1998.

    Google Scholar 

  41. Tzavaras, A.E.: Materials with internal variables and relaxation to conservation laws. Arch. Rational Mech. Anal. 146 (1999), 129–155.

    MathSciNet  MATH  Google Scholar 

  42. Yang, T., Zhu, C. and H. Zhao: Compactness framework of LP approximate solutions for scalar conservation laws. J. Math. Anal. Appl. 220 (1998), 164–186.

    Article  MathSciNet  MATH  Google Scholar 

  43. DiPerna, R.J.: Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), 27–70.

    MathSciNet  MATH  Google Scholar 

  44. Greenberg, J.M.: Smooth and time periodic solutions to the quasilinear wave equation. Arch. Rational Mech. Anal. 60 (1975), 29–50.

    MATH  Google Scholar 

  45. Greenberg, J.M. and M. Rascle: Time-periodic solutions to systems of conservation laws. Arch. Rational Mech. Anal. 115 (1991), 395–407.

    MathSciNet  MATH  Google Scholar 

  46. Chen, G.-Q.: Remarks on global solutions to the compressible Euler equations with spherical symmetry. Proc. Royal Soc. Edinburgh 127A (1997), 243–259.

    Article  Google Scholar 

  47. Chen, G.-Q. and J. Glimm: Global solutions to the compressible Euler equations with geometric structure. Comm. Math. Phys. 180 (1996), 153–193.

    Article  MathSciNet  MATH  Google Scholar 

  48. Chen, G.-Q. and J. Glimm: Global solutions to the cylindrically symmetric rotating motion of isentropic gas. ZAMP 47 (1996), 353–372.

    Google Scholar 

  49. Chen, G.-Q. and P.-T. Kan: Hyperbolic conservation laws with umbilic degeneracy, L Arch. Rational Mech. Anal. 130 (1995), 231–276.

    MathSciNet  MATH  Google Scholar 

  50. Heidrich, A.: Global weak solutions to initial-boundary value problems for the one-dimensional quasi-linear wave equation with large data. Arch. Rational Mech. Anal. 126 (1994), 333–368.

    MathSciNet  MATH  Google Scholar 

  51. Kan, P.T., Santos, M.M. and Z. Xin: Initial-boundary value problem for conservation laws. Comm. Math. Phys. 186 (1997), 701–730.

    Article  MathSciNet  MATH  Google Scholar 

  52. Marcati, P. and R. Natalini: Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation. Arch. Rational Mech. Anal. 129 (1995), 129–145.

    MathSciNet  MATH  Google Scholar 

  53. Marcati, P. and B. Rubino: Hyperbolic to parabolic relaxation theory for quasilinear first order systems. (Preprint).

    Google Scholar 

  54. Yang, T., Zhu, C. and H. Zhao: Compactness framework of LP approximate solutions for scalar conservation laws. J. Math. Anal. Appl. 220 (1998), 164–186.

    Article  MathSciNet  MATH  Google Scholar 

  55. Dafermos, C.M.: Asymptotic behavior of solutions of a hyperbolic conservation law. J. Diff. Eqs. 11 (1972), 416–424.

    Article  MathSciNet  MATH  Google Scholar 

  56. Frid, H.: Initial-boundary value problems for conservation laws. J. Diff. Eqs. 128 (1996), 1–45.

    Article  MathSciNet  MATH  Google Scholar 

  57. Schaeffer D. and M. Shearer: The classification of 2 x 2 nonstrictly hyperbolic conservation laws, with application to oil recovery. Comm. Pure Appl. Math. 40 (1987), 141–178.

    MathSciNet  MATH  Google Scholar 

  58. Bonnefille, M.: Propagation des oscillations dans deux classes de systèmes hyperboliques (2 x 2 et 3 x 3). Comm. PDE 13 (1988), 905–925.

    Article  MathSciNet  MATH  Google Scholar 

  59. Chen, G.-Q.: Propagation and cancellation of oscillations for hyperbolic systems of conservation laws. Comm. Pure Appl. Math. 44 (1991), 121–139.

    MATH  Google Scholar 

  60. Chen, G.-Q.: Hyperbolic systems of conservation laws with a symmetry. Comm. PDE 16 (1991), 1461–1487.

    Article  MATH  Google Scholar 

  61. Chen, G.-Q.: The method of quasidecoupling for discontinuous solutions to conservation laws. Arch. Rational Mech. Anal. 121 (1992), 131–185.

    MATH  Google Scholar 

  62. Heibig, A.: Error estimates for solutions to hyperbolic systems of conservation laws. Comm. PDE 18 (1993), 281–304.

    Article  MathSciNet  MATH  Google Scholar 

  63. Poupaud, F., Rascle, M. and J.P. Vila: Global solutions to the isothermal Euler-Poisson system with arbitrarily large data. J. Diff. Eqs. 123 (1995), 93–121.

    Article  MathSciNet  MATH  Google Scholar 

  64. Landau, L.D.: On shock waves at large distances from their place of origin. J. Phys. USSR 9 (1945), 495–500.

    Google Scholar 

  65. Lighthill, M.J.: A method for rendering approximate solutions to physical problems uniformly valid. Philos. Magazine 40 (1949), 1179–1201.

    MathSciNet  MATH  Google Scholar 

  66. Whitham, G.B.: The flow pattern of a supersonic projectile. Comm. Pure Appl. Math. 5 (1952), 301348.

    Google Scholar 

  67. Choquet-Bruhat, V.: Ondes asymptotiques et approchées pour systèmes d’équations aux dérivées paratielles nonlinéaires. J. Math. Pures Appl. 48 (1969), 117–158.

    MathSciNet  MATH  Google Scholar 

  68. Hunter, J.K., and J.B. Keller: Weakly nonlinear high frequency waves. Comm. Pure Appl. Math. 36 (1983), 547–569.

    MathSciNet  MATH  Google Scholar 

  69. Hunter, J.K., and J.B. Keller: Nonlinear hyperbolic waves. Proc. Royal Soc. London 417A (1988), 299–308.

    MathSciNet  Google Scholar 

  70. Majda, A. and R. Pego: Stable viscosity matrices for systems of conservation laws. J. Diff. Eqs. 56 (1985), 229–262.

    Article  MathSciNet  MATH  Google Scholar 

  71. Majda, A., Rosales, R. and M. Schonbek: A canonical system of integrodifferential equations arising in resonant nonlinear acoustics. Studies Appl. Math. 79 (1988), 205–262.

    MathSciNet  MATH  Google Scholar 

  72. Pego, R.L.: Stable viscosities and shock profiles for systems of conservation laws. Trans. AMS 282 (1984), 749–763.

    Article  MathSciNet  MATH  Google Scholar 

  73. Hunter, J.: Interaction of elastic waves. Stud. Appl. Math. 86 (1992), 281–314.

    MathSciNet  MATH  Google Scholar 

  74. Joly, J.-L., Métivier, G. and J. Rauch: Resonant one-dimensional nonlinear geometric optics. J. Funct. Anal. 114 (1993), 106–231.

    Article  MathSciNet  Google Scholar 

  75. Joly, J.-L., Métivier, G. and J. Rauch: Coherent and focusing multi-dimensional nonlinear geometric optics. Ann. Sc1. ENS 28 (1995), 51–113.

    MATH  Google Scholar 

  76. Cheverry, C.: The modulation equations of nonlinear geometric optics. Comm. PDE 21 (1996), 1119–1140.

    Article  MathSciNet  MATH  Google Scholar 

  77. Crandall, M.G. and A. Majda: The method of fractional steps for conservation laws. Math. Comput. 34 (1980), 285–314.

    MATH  Google Scholar 

  78. Schochet, S.: Resonant nonlinear geometric optics for weak solutions of conservation laws. J. Diff. Eqs. 113 (1994), 473–504.

    Article  MathSciNet  MATH  Google Scholar 

  79. Cheverry, C.: Justification de l’optique géométrique non linéaire pour un système de lois de conservations. Duke Math. J. 87 (1997), 213–263.

    MathSciNet  MATH  Google Scholar 

  80. Majda, A.: Nonlinear geometric optics for hyperbolic systems of conservation laws. Oscillation Theory, Computation and Methods of Compensated Compactness, pp. 115–165, eds. C. Dafermos, J.L. Ericksen, D. Kinderlehrer and M. Slemrod. New York: Springer, 1986.

    Google Scholar 

  81. Serre, D.: Systèmes de Lois de Conservation, Vols. I-I1. Paris: Diderot, 1996. English translation: Systems of Conservation Laws, Vols. 1–2. Cambridge: Cambridge University Press, 1999.

    Google Scholar 

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  1. Division of Applied Mathematics, Brown University, 02912, Providence, RI, USA

    Constantine M. Dafermos

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Dafermos, C.M. (2000). Compensated Compactness. In: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wissenschaften, vol 325. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-22019-1_15

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  • DOI: https://doi.org/10.1007/978-3-662-22019-1_15

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