Advertisement

Analysis in Classes of Discontinuous Functions and Partial Differential Equations

  • A. I. Volpert
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 98)

Abstract

Review of old and new results in analysis in classes of functions whose generalized derivatives are measures and its applications to partial differential equations and continuum mechanics is given.

Keywords

Hyperbolic System Regular Point Generalize Derivative Entropy Condition Discontinuous Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Da1]
    C.M. Dafermos, Admissible wave fans in nonlinear hyperbolic systems, Arch. Rat. Mech. Anal. 106 (1989), 243–260.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [Da2]
    C.M. Dafermos, Generalized characteristics in hyperbolic systems of conservation laws, Arch. Rat Mech. Anal. 107 (1989), 127–155.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [DG1]
    E. 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.CrossRefGoogle Scholar
  4. [DG2]
    E. De Giorgi, Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio ad r dimensioni, Ricerche Mat. 4(1955), 95–113.Google Scholar
  5. [DM]
    R.J. Di Perna and A. Maida, The validity of nonlinear geometric optics for weak solutions of conservation laws, Comm. Math. Phys. 98 (1985), 313–347.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [DP1]
    R.J. Di Perna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws. Ind. Univ. Math. J. 24 (1975), 1047–1071.CrossRefGoogle Scholar
  7. [DP2]
    R.J. Di Perna, Uniqueness of solutions of hyperbolic systems of conservation laws, Ind. Univ. Math. J. 28 (1979), 137–188.CrossRefGoogle Scholar
  8. [DP3]
    R.J. Di Perna, Convergence of approximate solutions to conservation laws, Arch. Rat Mech. Anal. 82 (1983), 27–70.CrossRefGoogle Scholar
  9. [FL]
    K.O. Friedrichs and P.D. Lax, Systems of conservation equations with a convex extension, Proc. Nat Acad. Sci. USA 68 (1971), 1636–1688.MathSciNetCrossRefGoogle Scholar
  10. [FR]
    W.H. Fleming and R. Rishel, An integral formula for total gradient variation, Arch. Math. 11 (1960), 218–222.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [F1]
    H. Federer, An analytic characterization of distributions whose partial derivatives are representable by measures, Bull Amer. Math. Soc. 60 (1954), 339.Google Scholar
  12. [F2]
    H. Federer, The Gauss-Green theorem, Trans. Amer. Math. Soc. 58 (1945), 44–76.MathSciNetzbMATHGoogle Scholar
  13. [F3]
    H. Federer, A note on the Gauss-Green theorem, Proc. Amer. Math. Soc. 9 (1958), 447–451.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [G]
    S.K. Godunov, The problem of a generalized solutions in the theory of quasi-linear equations and in gas dynamics, Russian Math. Surveys 17 (1962), 145–156.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [G1]
    J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [GW]
    C. Goffman and D. Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc. 12 (1961), 116–121.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [H]
    A. Harten, On the symmetric form of systems of conservation laws with entropy, Journ. Comp. Physics 49 (1983), 151–164.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [HL]
    T.Y. Hou and Ph. Le Floch, Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comp. 62 No.206 (1994), 497–530.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [K]
    K. Krickeberg, Distributionen, Funktionen bestrankter Variation and Lebesguescher Inhalt nichtparametrischer Flachen, Ann. Math. Pura Appl. (4) 44 (1957), 105–133.MathSciNetGoogle Scholar
  20. [Kr]
    S.N. Kruzkov, First-order quasi-linear equations in several independent variables, Math. USSR Sbornik 10 (1970), 127–243.CrossRefGoogle Scholar
  21. [L]
    P.D. Lax, Shock waves and entropy “Contributions to Nonlinear Functional Analysis”, 603-634, Academic Press, New York, 1971.Google Scholar
  22. [LF]
    Ph. Le Floch, Entropy weak solutions to nonlinear hyperbolic systems under non-conservative form, Comm. Part. Dif. Eq. 13 (1988), 669–727.zbMATHCrossRefGoogle Scholar
  23. [Li]
    T.P. Liu, Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws, C.P.A.M. 30 (1977), 585–610.Google Scholar
  24. [LL]
    Ph. Le Floch and Tai-Ping Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 No.3 (1993), 261–280.MathSciNetzbMATHGoogle Scholar
  25. [S]
    S. Schochet, Resonant nonlinear geometric optics for weak solutions of conservation laws, J. Dif. Equat. 113 (1994), 473–504.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [T]
    E. Tadmor, Entropy functions for symmetric systems of conservation laws, Journ. Math. Anal Appl 122 (1987), 355–359.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [Tr]
    C. Truesdell, A first course in rational continuum mechanics, 1972.Google Scholar
  28. [VD]
    A.I. Volpert and S.I. Doronin, On numerical methods for nonconservative hyperbolic equations, Preprint, ICPh. (1992).Google Scholar
  29. [VH1]
    A.I. Volpert and S.I. Hudjaev, Analysis in classes of discontinuous functions and equations of mathematical physics, Martinus Nijhoff Publishers, 1985, 678p.Google Scholar
  30. [VH2]
    A.I. Volpert and S.I. Hudjaev, Cauchy problem for degenarate second order quasi-linear parabolic equations, Math. USSR Sbornik 7 (1969), 365–387.CrossRefGoogle Scholar
  31. [VT1]
    A.I. Volpert and R.S. Tishakova, Positive solutions of quasi-linear parabolic system of equations, Preprint, Chernogolovka, 1980, 20p. (In Russian).Google Scholar
  32. [VT2]
    A.I.Volpert and R.S. Tishakova, Positive solutions of the second boundary value problem for quasi-linear parabolic systems, Preprint, Chernogolovka, 1981, 13p. (In Russian).Google Scholar
  33. [VZ]
    A.I. Volpert, A.M. Zhiljaev, V.P. Filipenko, Mathematical problems of detonation waves stability, Proceedings of 9 Symp. Comb. Explosion, Detonation (1989), 6-9. (In Russian).Google Scholar
  34. [V1]
    A.I. Volpert, The spaces BV and quasi-linear equations, Math. USSR Sbornik 2 No.2 (1967), 225–267.CrossRefGoogle Scholar
  35. [V2]
    A.I. Volpert, BV-Analysis and hydrodynamics, Preprint, Chernogolovka, 1991, 32p.Google Scholar

Copyright information

© Springer Basel AG 1997

Authors and Affiliations

  • A. I. Volpert
    • 1
  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

Personalised recommendations