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Unsteady flow in porous media with a free surface

  • Hideo Kawarada
  • Hideyuki Koshigoe
Article

Abstract

The short time existence of the unsteady free boundary appearing in the porous media is discussed by use of Nash-Moser's implicit function theorem. The uniqueness, existence and the semibounded estimates for the solution of the linearized equation play an important role in the proof of the existence theorem. Also Nash-Moser's implicit function theorem is modified in an applicable form.

Key words

porous media free boundary Newton's method linearized equation evolution equation 

References

  1. [1]
    M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications. Dekker, New York-Basel, 1986.zbMATHGoogle Scholar
  2. [2]
    H.W. Alt and E. Di Benedetto, Nonsteady flow of water and oil through inhomogeneous porous media. Ann. Scuola Norm. Sup. Pisa, Cl. Sci. IV,XII (1985), 335–392.Google Scholar
  3. [3]
    H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z.,183 (1983), 311–341.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Dervieux, Perturbation des equations d'equilibre dun plasma confine. INRIA Report, 1980.Google Scholar
  5. [5]
    E. Dibenedetto and A. Friedman, Periodic behaviour for the evolutionary dam problem and related free boundary problems. Comm. Partial Differential Equations11(12), (1986), 1297–1377.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    A. Friedman, Partial Differential Equation. Kreger, Robert E. Krieger, Huntington, New York, 1976.Google Scholar
  7. [7]
    D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg-New York, 1977.zbMATHGoogle Scholar
  8. [8]
    J. Hadamard, Mémoire sur le problème d'analyse relatif a l'équilibre des plaques élastiques encastrées. Oeuvre de Jacques Hadamard, C.N.R.S., Paris, 1968.Google Scholar
  9. [9]
    T. Kato, Linear evolution equations of “hyperbolic” type I. J. Fac. Sci., Univ. Tokyo Sect. IA Math.,17 (1970), 241–258.zbMATHGoogle Scholar
  10. [10]
    T. Kato, Linear evolution equations of “hyperbolic” type, II. J. Math. Soc. Japan,25 (1973), 648–666.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    H. Kawarada and H. Koshigoe, Nonlinear evolution equations of Moser's type, II. Tech. Rep. Math. Sci. Chiba Univ.,14 (1987), 1–37.Google Scholar
  12. [12]
    H. Kawarada and H. Koshigoe, Modified Nash-Moser's implicit function theorem and its application to nonlinear wave equations (to appear).Google Scholar
  13. [13]
    J. Moser, A rapidly convergent iteration method and nonlinear partial differential equations I. Ann. Scuola Norm. Sup. Pisa,20 (1966), 265–315.MathSciNetGoogle Scholar
  14. [14]
    J. Nash, The embedding problem for Riemannian manifolds. Ann. of Math.63 (1956) 20–63.CrossRefMathSciNetGoogle Scholar
  15. [15]
    H. Rasmussen and D. Salhani, Unsteady porous flow with a free surface. IMA J. Appl. Math.,27 (1981), 307–318.zbMATHCrossRefGoogle Scholar
  16. [16]
    M. Todsen, On the solution of transient free surface flow problems in porous media by finite-difference methods. J. Hydrology12 (1971), 177–210.CrossRefGoogle Scholar
  17. [17]
    H. Tanabe, Equations of Evolutions. North Holland, Amsterdam-New York-Oxford; 1984.Google Scholar
  18. [18]
    H. Kawarada and H. Koshigoe, The unsteady flow in the porous media with a free surface (in Japanese). Proceedings of 1988 joint meeting of applied mathematics (RIMS., Kyoto Univ.), 11–15.Google Scholar

Copyright information

© JJIAM Publishing Committee 1991

Authors and Affiliations

  • Hideo Kawarada
    • 1
  • Hideyuki Koshigoe
    • 1
  1. 1.Institute of Applied MathematicsChiba UniversityChibaJapan

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