Advertisement

First-Order Evolution Equations and the Galerkin Method

  • Eberhard Zeidler

Abstract

In this chapter, we generalize the Hilbert space methods in Chapter 23, for the investigation of linear parabolic differential equations, to nonlinear problems. In this connection, as an essential auxiliary tool, we use the Galerkin method.

Keywords

Galerkin Method Theta Function Dual Pair Nonlinear Evolution Equation Schrodinger Equation 
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 to the Literature

  1. Caroll, R. (1969): Abstract Methods in Partial Differential Equations. Harper and Row, New York.Google Scholar
  2. Gajewski, H., Gröger, K., and Zacharias, K. (1974): Nichtlineare Operator gleichungen und Operator differ entialgleichungen. Akademie-Verlag, Berlin.Google Scholar
  3. Barbu, V. (1976): Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden.CrossRefMATHGoogle Scholar
  4. Tanabe, H. (1979): Equations of Evolution. Pitman, London.MATHGoogle Scholar
  5. Henry, D. (1981): Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Vol. 840. Springer-Verlag, Berlin.Google Scholar
  6. Haraux, A. (1981): Nonlinear Evolution Equations: Global Behavior of Solutions. Lecture Notes in Mathematics, Vol. 841. Springer-Verlag, Berlin.Google Scholar
  7. Višik, M. and Fursikov, A. (1981): Mathematical Problems in Statistical Hydromechanics. Nauka, Moscow (Russian). (German edition: Mathematische Probleme der statistischen Hydromechanik. Teubner, Leipzig, 1986.)Google Scholar
  8. Pazy, A. (1983): Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  9. Smoller, J. (1983): Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  10. Majda, A. (1984): Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  11. Wahl, W. v. (1985): The Equations of Navier-Stokes and Abstract Parabolic Equations. Vieweg, Braunschweig.Google Scholar
  12. Browder, F. [ed.] (1986): Nonlinear Functional Analysis and Its Applications. Proc. Sympos. Pure Math., Vol. 45, Parts 1, 2. Amer. Math. Soc., Providence, RI.Google Scholar
  13. Pavel, N. (1987): Nonlinear Evolution Operators and Semigroups. Springer-Verlag, New York.MATHGoogle Scholar
  14. Benilan, P., Crandall, M., and Pazy, A. (1989): Nonlinear Evolution Governed by Accretive Operators (monograph to appear).Google Scholar
  15. Christodoulou, D. and Klainerman, S. (1990): Nonlinear Hyperbolic Equations (monograph to appear).Google Scholar
  16. Dubinskii, Ju. (1968): Quasilinear elliptic and parabolic equations of arbitrary order. Uspekhi Mat. Nauk 23 (139), 45–90 (Russian).Google Scholar
  17. Dubinskii, Ju. (1976): Nonlinear elliptic and parabolic equations. Itogi Nauki i Tekhniki, Sovremennye problemy matematikii 9, 1–130 (Russian).Google Scholar
  18. Wahl, W. v. (1978): Uber die stationaren Gleichungen von Navier-Stokes, semilineare elliptische und parabolische Gleichungen. Jahresber. Deutsch. Math.-Verein. 80, 129–149.Google Scholar
  19. Wahl, W. v. (1982): Nichtlineare Evolutionsgleichungen. In: Kurke, H. et al. [eds.], Recent Trends in Mathematics, pp. 294–302. Teubner, Leipzig.Google Scholar
  20. Kato, T. (1986): Nonlinear equations of evolution in B-spaces. In: Browder, F. [ed.] (1986), Part 2, pp. 9–24.Google Scholar
  21. Brezis, H., Crandall, M., and Kappel, F. [eds.] (1986): Semigroups and Applications, Vols. 1, 2. Wiley, New York.MATHGoogle Scholar
  22. Kato, T. and Lai, C. (1984): Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56, 15–28.Google Scholar
  23. Kato, T. (1986): Nonlinear equations of evolution in B-spaces. In: Browder, F. [ed.] (1986), Part 2, pp. 9–24.Google Scholar
  24. Oleinik, O. and Kruzkov, S. (1961): Quasilinear parabolic equations of second order. Uspekhi Mat. Nauk 16 (5), 115–155 (Russian).Google Scholar
  25. Visik, M. (1962): On initial-boundary value problems for quasilinear parabolic equations of higher order. Mat. Sbornik 59, 289–325 (Russian).Google Scholar
  26. Ladyzenskaja, O., Solonnikov, V., and Uralceva, N. (1967): Linear and Quasilinear Parabolic Equations. Nauka, Moscow (Russian).Google Scholar
  27. Friedman, A. (1964): Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  28. Friedman, A. (1969): Partial Differential Equations. Holt, Rinehart, and Winston, New York. (Second edition 1976.)Google Scholar
  29. Lions, J. (1969): Quelques methodes de resolution des problemes aux limites non lineaires. Dunod, Paris.MATHGoogle Scholar
  30. Dubinskii, Ju. (1968): Quasilinear elliptic and parabolic equations of arbitrary order. Uspekhi Mat. Nauk 23 (139), 45–90 (Russian).Google Scholar
  31. Dubinskii, Ju. (1976): Nonlinear elliptic and parabolic equations. Itogi Nauki i Tekhniki, Sovremennye problemy matematikii 9, 1–130 (Russian).Google Scholar
  32. Amann, H. (1986): Quasilinear evolution equations and parabolic systems. Trans. Amer. Math. Soc. 293, 191–227.Google Scholar
  33. Amann, H. (1986a): Quasilinear parabolic systems under nonlinear boundary conditions. Arch. Rational Mech. Anal. 92, 153–192.Google Scholar
  34. Amann, H. (1986b): Semigroups and nonlinear evolution equations. Linear Algebra Appl. 84, 3–32.Google Scholar
  35. Amann, H. (1986c): Parabolic evolution equations with nonlinear boundary conditions. In: Browder, F. [ed.] (1986), Part 1, pp. 17–27.Google Scholar
  36. Amann, H. (1988): Remarks on quasilinear parabolic systems (to appear).Google Scholar
  37. Amann, H. (1988a): Dynamic theory of quasilinear parabolic equations, I, II (to appear).Google Scholar
  38. Amann, H. (1988b): Parabolic evolution equations in interpolation and extrapolation spaces. J. Funct. Anal. 78, 233–277.Google Scholar
  39. Amann, H. (1988c): Parabolic equations and nonlinear boundary conditions. J. Differential Equations 72, 201–269.Google Scholar
  40. Fife, P. (1979): Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomathematics, Vol. 28. Springer-Verlag, Berlin.Google Scholar
  41. Smoller, J. (1983): Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  42. Rothe, F. (1984): Global Solutions of Reaction-Diffusion Systems. Springer-Verlag, New York.MATHGoogle Scholar
  43. Amann, H. (1984): Existence and regularity for semilinear parabolic evolution equations. Ann. Scuola Norm. Sup. Pisa, CI. Sci. Serie IV, 11, 593–676.Google Scholar
  44. Amann, H. (1985): Global existence for semilinear parabolic systems. J. Reine. Angew. Math. 360, 47–83.Google Scholar
  45. Amann, H. (1986c): Parabolic evolution equations with nonlinear boundary conditions. In: Browder, F. [ed.] (1986), Part 1, pp. 17–27.Google Scholar
  46. Amann, H. (1988): Remarks on quasilinear parabolic systems (to appear).Google Scholar
  47. Amann, H. (1988a): Dynamic theory of quasilinear parabolic equations, I, II (to appear).Google Scholar
  48. Browder, F. [ed.] (1986): Nonlinear Functional Analysis and Its Applications. Proc. Sympos. Pure Math., Vol. 45, Parts 1, 2. Amer. Math. Soc., Providence, RI.Google Scholar
  49. Pao, C. (1978): Asymptotic behavior and nonexistence of global solutions for a class of nonlinear boundary value problems of parabolic type. J. Math. Anal. Appl. 65, 616–637.Google Scholar
  50. Pao, C. (1979): Bifurcation analysis on a nonlinear diffusion system in reactor dynamics. Applicable Anal. 9, 107–119.Google Scholar
  51. Pao, C. (1980): Nonexistence of global solutions for an integro-differential system in reactor dynamics. SI AM J. Math. Anal. 11, 559–564.Google Scholar
  52. Gajewski, H., Groger, K., and Zacharias, K. (1974): Nichtlineare Operator gleichungen und Operator differ entialgleichungen. Akademie-Verlag, Berlin.Google Scholar
  53. Caroll, R. and Showaiter, R. (1976): Singular and Degenerate Cauchy Problems. Academic Press, New York.Google Scholar
  54. Bohm, M. and Showalter, R. (1985): A nonlinear pseudoparabolic diffusion equation. SIAM J. Math. Anal. 16, 980–999.Google Scholar
  55. Kacur, J. (1985): Method of Rothe in Evolution Equations. Teubner, Leipzig.MATHGoogle Scholar
  56. Nirenberg, L. (1972): An abstract form of the nonlinear Cauchy-Kovalevskaja theorem. J. Differential Geom. 6, 561–576.Google Scholar
  57. Ovsjannikov, L. (1976): Cauchy problem in a scale of Banach spaces and its application to the shallow water justification. In: Germain, P. and Nayroles, B. [eds.] (1976), pp. 426–437.Google Scholar
  58. Nishida, T. (1977): On a theorem of Nirenberg. J. Differential Geom. 12, 629–633.Google Scholar
  59. Hamilton, R. (1982): The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7, 65–222.Google Scholar
  60. Deimling, K. (1985): Nonlinear Functional Analysis. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  61. Walter, W. (1985): Functional differential equations of the Cauchy-Kovalevskaja type. Aequationes Math. 28, 102–103.Google Scholar
  62. Novikov, S. et al. (1980): Theory of Solitons. Nauka, Moscow (Russian). (English edition: Plenum, New York, 1984.)Google Scholar
  63. Ablowitz, M. and Segur, H. (1981): Solitons and the Inverse Scattering Transform. SIAM, Philadelphia, PA.CrossRefMATHGoogle Scholar
  64. Gardner, C., Green, J., Kruskal, M., and Miura, R. (1967): Method for solving the Korteweg-de Vries equation. Phys. Lett. 19, 1095–1097.Google Scholar
  65. Lax, P. (1968): Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467–490.Google Scholar
  66. Bullough, R. and Caudrey, P. [eds.] (1980): Solitons. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  67. Calogero, F. and Degasperis, A. (1982): Spectral Transform and Solitons. North- Holland, Amsterdam.MATHGoogle Scholar
  68. Lee, T. (1981): Particle Physics and Introduction to Field Theory. Harwood, New York.Google Scholar
  69. Rajaraman, R. (1982): Solitons and Instantons. North-Holland, Amsterdam.MATHGoogle Scholar
  70. Davydov, A. (1984): Solitons in Molecular Systems. Reidel, Boston, MA.Google Scholar
  71. Knorrer, H. (1986): Integrable Hamiltonsche Systeme und algebraische Geometrie. Jahresber. Deutsch. Math.-Verein. 88, 82–103.Google Scholar
  72. Faddeev, L. and Takhtadjan, L. (1987): The Hamiltonian Approach to the Theory of Solitons. Springer-Verlag, New York.CrossRefGoogle Scholar
  73. Bobenko, A. and Bordag, L. (1987): The qualitative analysis of multi-cnoidal waves of the Korteweg-de Vries equation via automorphic functions. Zapiski Naucnych Seminarov LOMI, Vol. 165, pp. 31–41. Nauka, Leningrad (Russian). (English translation in J. Sov. Math, (to appear). )Google Scholar
  74. Konopelchenko, B. (1987): Nonlinear Integrable Equations. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  75. Newell, A. et al. (1987): Soliton Mathematics. Les Presses de l’Universite de Montreal.Google Scholar
  76. Toda, M. (1981): Theory of Nonlinear Lattices. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  77. Its, A. and Matveev, V. (1975): Schrodinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation. Theoret. Math. Phys. 23 (1), 51–67 (Russian).Google Scholar
  78. Matveev, V. et al (1986): Algebraic-geometrical principles of the superposition of finite- gap solutions of integrable nonlinear equations. Uspekhi Mat. Nauk 41 (2), 3–42 (Russian).Google Scholar
  79. Bobenko, A., Its, A., and Matveev, V. (1991): Analytic Theory of Solitons. Nauka, Moscow (Russian) (monograph to appear).Google Scholar
  80. Mumford, D. (1983): Tata Lectures on Theta. Birkhauser, Boston, MA.Google Scholar
  81. Mihailov, A., Sabat, A., and Jamilov, R. (1987): A symmetric approach to the classification of nonlinear equations: A full classification of integrable systems. Uspekhi Mat. Nauk 42 (4), 3–53 (Russian).Google Scholar
  82. Atiyah, M. and Hitchin, N. (1988): The Geometry and Dynamics of Magnetic Monopoles. University Press, Princeton, NJ.Google Scholar
  83. Brebbia, C. et al. (1984): Boundary Element Techniques: Theory and Applications in Engineering. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  84. Schechter, M. (1981): Operator Methods in Quantum Mechanics. North-Holland, Amsterdam.MATHGoogle Scholar
  85. Reed, M. and Simon, B. (1971): Methods of Modern Mathematical Physics, Vols. 1–4. Academic Press, 1971.Google Scholar
  86. Berezin, F. and Shubin, M. (1983): The Schrodinger Equation. University Press, Moscow (Russian).Google Scholar
  87. Poschel, J. and Trubowitz, E. (1987): Inverse Spectral Theory. Academic Press, New York.Google Scholar
  88. Bona, J. and Smith, R. (1975): The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London A278, 555–601.Google Scholar
  89. Bona, J. and Scott, R. (1976): Solution of the Korteweg-de Vries equation in fractional order Sobolev spaces. Duke Math. J. 43, 87–99.Google Scholar
  90. Kato, T. (1983): On the Cauchy problem for the generalized Korteweg-de Vries equations. In: Guillemin, V. [ed.] (1983), Studies in Applied Mathematics, Vol. 8, pp. 93–128. Academic Press, New York.Google Scholar
  91. Kato, T. (1986): Nonlinear equations of evolution in B-spaces. In: Browder, F. [ed.] (1986), Part 2, pp. 9–24.Google Scholar
  92. Kruzkov, S. and Faminskii, A. (1985): Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation. Math. USSR-Sbornik 48, 391–421.Google Scholar
  93. Ciarlet, P. and Lions, J. [eds.] (1988): Handbook of Numerical Analysis. Vol. 1: Finite Element Method. Vol 2: Finite Difference Method (Vols. 3ff to appear). North- Holland, Amsterdam.Google Scholar
  94. Thomee, V. (1984): Galerkin Finite Element Methods for Parabolic Problems. Lecture Notes in Mathematics, Vol. 1054. Springer-Verlag, Berlin.Google Scholar
  95. Douglas, J. and Dupont, T. (1970): Galerkin methods for parabolic equations. SI AM J. Numer. Anal. 7, 575–626.Google Scholar
  96. Glowinski, R., Lions, J., and Tremolieres, R. (1976): Analyse numerique des inequations variationelles, Vols. 1,2. Gauthier-Villars, Paris. (English edition: North-Holland, Amsterdam, 1981.)Google Scholar
  97. Ciarlet, P. (1977): Numerical Analysis of the Finite Element Method for Elliptic Boundary Value Problems. North-Holland, Amsterdam.Google Scholar
  98. Temam, R. (1977): Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam. (Third revised edition, 1984.)Google Scholar
  99. Fletcher, C. (1984): Computational Galerkin Methods. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  100. Gekeler, E. (1984): Discretization Methods for Stable Initial-Value Problems. Lecture Notes in Mathematics, Vol. 1044. Springer-Verlag, Berlin.Google Scholar
  101. Reinhardt, H. (1985): Analysis of Approximation Methods for Differential- and Integral Equations. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  102. Girault, V. and Raviart, P. (1986): Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, New York.CrossRefGoogle Scholar
  103. Raviart, P. (1967): Sur V approximation de certaines equations cf evolution lineaires. J. Math. Pures Appl. 46, 11–183.Google Scholar
  104. Janenko, N. (1969): Die Zwischenschrittmethode zur Losung mehrdimensionaler Prob- leme der mathematischen Physik. Lecture Notes in Mathematics, Vol. 91. Springer- Verlag, Berlin.Google Scholar
  105. Temam, R. (1968): Sur la stabilite et la convergence de la method des pas fractionnaires. Ann. Mat. Pura Appl. 79, 191–379.Google Scholar
  106. Temam, R. (1977): Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam. (Third revised edition, 1984.)Google Scholar
  107. Lions, J. (1969): Quelques methodes de resolution des problemes aux limites non lineaires. Dunod, Paris.MATHGoogle Scholar
  108. Kacur, J. (1985): Method of Rothe in Evolution Equations. Teubner, Leipzig.MATHGoogle Scholar
  109. Schumann, R. (1987): The convergence of Rothe’s method for parabolic differential equations. Z. Anal. Anwendungen 6, 559–574.Google Scholar
  110. Meis, T. and Marcowitz, U. (1978): Numerische Behandlung partieller Differential- gleichungen. Springer-Verlag, Berlin. (English edition: Numerical Solutions of Partial Differential Equations. Springer-Verlag, New York, 1981.)Google Scholar
  111. Marchuk, G. and Shaidurov, V. (1983): Difference Methods and Their Extrapolations. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  112. Reinhardt, H. (1985): Analysis of Approximation Methods for Differential- and Integral Equations. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  113. Marchuk, G. and Shaidurov, V. (1983): Difference Methods and Their Extrapolations. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  114. Stiiben, K. and Trottenberg, U. (1982): Multigrid methods. In: Hackbusch, W. and Trottenberg, U. [eds.] (1982), pp. 1–176.Google Scholar
  115. Lichnewsky, A. and Saguez, C. [eds.] (1987): Super computing: the State of the Art. North-Holland, Amsterdam.Google Scholar
  116. Martin, J. [ed.] (1988): Performance Evaluation of Supercomputers. North-Holland, Amsterdam.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Eberhard Zeidler
    • 1
  1. 1.Sektion MathematikLeipzigGerman Democratic Republic

Personalised recommendations