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Maximal regularity for a free boundary problem

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Abstract

This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darcy's law.

We prove existence of a unique maximal classical solution, using methods from the theory of maximal regularity, analytic semigroups, and Fourier multipliers. Moreover, we describe a state space which can be considered as domain of parabolicity for the problem under consideration.

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References

  1. S. AGMON, A. DOUGLIS, L. NIRENBERG, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,Comm. Pure Appl. Math. 12, 623–727 (1959)

    Google Scholar 

  2. H. AMANN, Existence and regularity for semilinear parabolic evolution equations,Ann. Scuola Norm. Pisa, (4) XI, 593–676 (1984).

    Google Scholar 

  3. H. AMANN, Dynamic theory of quasilinear parabolic equations. I.Nonlinear Anal., TMA 12, 895–919 (1988)

    Google Scholar 

  4. H. AMANNLinear and Quasilinear Parabolic Problems, Book in preparation, 1994

  5. H. AMANN, J. ESCHER, Strongly continuous dual semigroups, to appear inAnn. Mat. Pura Appl.

  6. S. B. ANGENENT, Nonlinear analytic semiflows,Proc. Roy. Soc. Edinburgh 115 A, 91–107 (1990)

    Google Scholar 

  7. J. BEAR, Y. BACHMAT,Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publisher, Boston, 1990

    Google Scholar 

  8. F. E. BROWDER, On the spectral theory of elliptic operators I,Math. Ann. 142, 22–130 (1961)

    Google Scholar 

  9. G. DA PRATO, P. GRISVARD, Equations d'évolution abstraites nonlinéaires de type parabolique,Ann. Mat. Pura Appl., (4) 120, 329–396 (1979)

    Google Scholar 

  10. C. M. ELLIOTT, J. R. OCKENDON,Weak and Variational Methods for Moving Boundary Problems, Pitman, Boston, 1982

    Google Scholar 

  11. J. ESCHER, Nonlinear elliptic systems with dynamic boundary conditions,Math. Z. 210, 413–439 (1992)

    Google Scholar 

  12. J. ESCHER, The Dirichlet-Neumann operator on continuous functions,Ann. Scuola Norm. Pisa, (4) XXI, 235–266 (1994)

    Google Scholar 

  13. A. FRIEDMAN, B. HU, The Stefan problem with kinetic condition at the free boundary,Ann. Scuola Norm. Pisa, (4) XLIV, 87–111 (1992)

    Google Scholar 

  14. D. GILBARG, N.S. TRUDINGER,Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1977

    Google Scholar 

  15. G. GRUBB,Functional Calculus of Pseudo-Differential Boundary Problems, Birkhäuser-Verlag, Basel, 1986

    Google Scholar 

  16. T. HINTERMANN, Evolution equations with dynamic boundary conditions,Proc. Roy. Soc. Edinburgh 113 A, 43–60 (1989)

    Google Scholar 

  17. B. HUNT, Vertical recharge of unconfined aquifer.J. Hydr. Divn. 97, 1017–1030 (1971)

    Google Scholar 

  18. A. LUNARDI. On the local dynamical system associated to a fully nonlinear abstract parabolic equation. InNonl. Anal. and Appl., ed. Lakshmikantham, M. Dekker, New York, 319–326 (1987)

    Google Scholar 

  19. A. LUNARDI, Analyticity of the maximal solution of an abstract nonlinear parabolic equation,Nonlinear Anal. TMA,6, 503–521 (1982)

    Google Scholar 

  20. A. LUNARDI,Analytic Semigroups and Optimal Regularity in Parabolic Equations, Birkhäuser-Verlag, Basel, 1994

    Google Scholar 

  21. H. KAWARADA, H. KOSHIGOE, Unsteady flow in porous media with a free surface,Japan J. Indust. Appl. Math.,8, 41–82 (1991)

    Google Scholar 

  22. B. E. PETERSON,Introduction to Fourier Transform and Pseudo-Differential Operators, Pitman, Boston, London, 1983

    Google Scholar 

  23. M. H. PROTTER, H. F. WEINBERGER,Maximum Principles in Differential Equations, Springer-Verlag, Berlin, 1984

    Google Scholar 

  24. G. SIMONETT, Zentrumsmannigfaltigkeiten für quasilineare parabolische Gleichungen.Institut für angewandte Analysis und Stochastik, Report Nr. 2, Berlin, 1992

  25. G. SIMONETT, Quasilinear parabolic equations and semiflows. InEvolution Equations, Control Theory, and Biomathematics, Lecture Notes in Pure and Appl. Math. M. Dekker, New York, 523–536 (1994)

    Google Scholar 

  26. E. SINESTRARI, Continuous interpolation spaces and spatial regularity in nonlinear Volterra integrodifferential equations,J. Integral Equs. 5, 287–308 (1983)

    Google Scholar 

  27. E. M. STEIN,Singular Integrals and Differentiability Properties of Functions, University Press, Princeton, 1970

    Google Scholar 

  28. H. TRIEBEL,Theory of Function Spaces, Birkhäuser-Verlag, Basel, 1983

    Google Scholar 

  29. H. TRIEBEL,Theory of Function Spaces II, Birkhäuser-Verlag, Basel, 1992

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Basel, 4051, Basel, Switzerland

    Joachim Escher

  2. Department of Mathematics, Vanderbilt University, 37240, Nashville, TN, U.S.A.

    Gieri Simonett

Authors
  1. Joachim Escher
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  2. Gieri Simonett
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Supported by Schweizerischer Nationalfonds

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Escher, J., Simonett, G. Maximal regularity for a free boundary problem. NoDEA 2, 463–510 (1995). https://doi.org/10.1007/BF01210620

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  • DOI: https://doi.org/10.1007/BF01210620

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