Journal of Mathematical Sciences

, Volume 143, Issue 4, pp 3253–3292 | Cite as

Newton’s polygon method and the local solvability of free boundary problems

  • B. Grec
  • E. V. Radkevich


For some qualitatively new problems with free (unknown) boundary, conditions of local solvability (with respect to time) are investigated.


Cauchy Problem Free Boundary Model Problem Besov Space Free Boundary Problem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. A. Adams, Sobolev Spaces, Academic Press (1975).Google Scholar
  2. 2.
    S. C. Antonsev, C. R. Gonsalves, and A. M. Meirmanov, “Local existence of classical solutions to the well-posed Hele-Shaw problem,” Port. Math., 59, Fasc. 4, 435–452 (2002).MathSciNetGoogle Scholar
  3. 3.
    J. Bailly, “Local existence of classical solutions to first-order parabolic equations describing free boundaries,” Nonlinear Anal., 32, No. 5, 583–599 (1996).CrossRefMathSciNetGoogle Scholar
  4. 4.
    I. Brailovsky, A. Babchin, M. Frankel, and G. Sivashinsky, “Fingering instability in water-oil displacement,” to appear.Google Scholar
  5. 5.
    G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Theory of Unsteady Filtration of Liquids and Gases [in Russian], Nedra, Moscow (1984).Google Scholar
  6. 6.
    J. W. Cahn and J. E. Hillard, “Free energy of a nonuniform system. Pt. I. Interfacial free energy,” J. Chemical Physics, 28, No. 2, 258–267 (1958).CrossRefGoogle Scholar
  7. 7.
    W. Dreyer and W. H. Müller, “A study of the coarsening in tin/lead solders,” Internat. J. Solids Structures, 37, 3841–3871 (2000).MATHCrossRefGoogle Scholar
  8. 8.
    R. Denk and L. R. Volevich, “A new class of parabolic problems connected with the Newton polygon,” to appear.Google Scholar
  9. 9.
    W. Dreyer and B. Wagner, Sharp-Interface Model for Eutectic Alloys. Pt. I. Concentration Dependent Surface Tension, Preprint N ISSN 0946-8633, Weierstrass-Institute fur Angewandte Analysis und Stochastik (2003).Google Scholar
  10. 10.
    J. Escher, J. Prüss, and G. Simonett, “Analytic solutions for a Stefan problem with Gibbs-Thomson correction,” J. Reine Angew. Math., 563, 1–52 (2003).MATHMathSciNetGoogle Scholar
  11. 11.
    E. V. Frolova, “Estimates for solutions to model problems connected with quasistationary approximation for the Stefan problem,” J. Math. Sci., 124, No. 3, 5054–5069 (2004).CrossRefMathSciNetGoogle Scholar
  12. 12.
    K. K. Golovkin and V. A. Solonnikov, “On an estimate of the convolution operators,” Zap. Nauch. Sem. LOMI, 6–86 (1988).Google Scholar
  13. 13.
    S. G. Gindikin and L. R. Volevich, The Method of Newton’s Polyhedron in the Theory of Partial Differential Equations, Math. Appl. (Soviet Ser.), Vol. 86, Kluwer Academic (1992).Google Scholar
  14. 14.
    E.-I. Hanzawa, “Classical solution of the Stefan problem,” Tohoku Math. J., 33, No. 2, 297–335 (1981).MATHMathSciNetGoogle Scholar
  15. 15.
    H. Heuser, Lehrbuch der Analysis, Teil 2, B. G. Teubner, Stuttgart (1981).MATHGoogle Scholar
  16. 16.
    L. Hörmander, Linear Partial Differential Operators, Mir, Moscow (1965).MATHGoogle Scholar
  17. 17.
    P. L. Lizorkin, “Generalized Liouville differentiation and functional spaces L (r)p,” Mat. Sb., 60, No. 3, 325–353 (1965).MathSciNetGoogle Scholar
  18. 18.
    O. A. Ladyzhenskaya, Boundary Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).Google Scholar
  19. 19.
    O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1964).Google Scholar
  20. 20.
    S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer (1975).Google Scholar
  21. 21.
    L. Nirenberg, Topics on Nonlinear Functional Analysis, New York Courant Inst. Math. Sciences. VIII (1974).Google Scholar
  22. 22.
    E. V. Radkevich, “The Gibbs-Thomson effect and existence conditions of a classical solution to the modified Stefan problem,” in: J. Chadam and H. Rasmussen, Free Boundary Problem Involving Solids, Pitman Research Notes in Math. Ser., Vol. 281, Longman Scientific and Technical (1992), pp. 135–142.Google Scholar
  23. 23.
    V. A. Solonnikov, “Estimates of the solution of some noncoercive initial boundary-value problems using theorems on multipliers in Fourier-Laplace integrals,” Zap. Nauch. Sem. LOMI, 220–227 (1987).Google Scholar
  24. 24.
    V. Visintin, Models of Phase Transitions, Birkhäuser, Boston (1996).MATHGoogle Scholar
  25. 25.
    L. R. Volevich, “Newton polygon and general parameter-elliptic (parabolic) systems,” Russ. J. Math. Phys., 8, 375–400 (2001).MATHMathSciNetGoogle Scholar
  26. 26.
    L. V. Volevich, “Solvability of boundary problems for the general elliptic system,” Mat. Sb., 68, No. 3, 373–416 (1965).MathSciNetGoogle Scholar
  27. 27.
    V. N. Vidorovich, A. E. Volpian, and G. M. Kurdiumov, Crystallization Direction and Physico-Chemical Analysis [in Russian], Khimia, Moscow (1976).Google Scholar
  28. 28.
    S. J. Watson, F. Otto, B. Y. Rubinstein, and S. H. Davis, “Coarsening dynamics for the convective Cahn-Hilliard equation,” to appear.Google Scholar
  29. 29.
    M. Zakhrchenko and E. Radkevich, “On the singular-limit problem of the extended Cahn-Hilliard model,” Tr. Sem. Petrovsk., 23, 309–336 (2004).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • B. Grec
  • E. V. Radkevich

There are no affiliations available

Personalised recommendations