Journal of Mathematical Sciences

, Volume 83, Issue 1, pp 22–37 | Cite as

Regularity for doubly nonlinear parabolic equations

  • A. V. Ivanov
Article

Abstract

Hölder estimates and the existence of Hölder continuous generalized solutions of the first boundary problem for doubly nonlinear studies of the turbulent filtration of a liquid or a gas through a porous medium are obtained. Bibliography: 47 titles.

Keywords

Filtration Porous Medium Generalize Solution Parabolic Equation Boundary Problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    A. S. Kalashnicov, “Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations,”Russian Math. Surveys,42, 169–222 (1987).CrossRefGoogle Scholar
  2. 2.
    L. S. Leibenson, “General problem of the movement of a compressible fluid in a porous medium,”Izv. Akad. Nauk SSSR, Geography and Geophysics,9, 7–10 (1945).MATHMathSciNetGoogle Scholar
  3. 3.
    J. I. Diaz and F. de Tellin, “On doubly nonlinear parabolic equation arising in some models related to turbulent regimes,” Preprint (1991).Google Scholar
  4. 4.
    P. A. Raviart, “Sur la resolution de certaines equations paraboliques non lineaires,”J. funct. Anal.,5, 299–328 (1970).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    J.-L. Lions,Quelques Methodes de Resolution de Problemes aux Limites non Lineaires, Dunod, Paris (1969).MATHGoogle Scholar
  6. 6.
    A. Bamberger, “Etude d'une equation doublement non lineaire,”J. Funct. Anal.,24 148–154 (1977).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    O. Grange and F. Mignot, “Sur la resolution d'une equation et d'une inequation paraboliques non lineairs,”J. Funct. Anal.,11, 77–92 (1972).CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    T. Arai, “On the existence of solutions for doubly nonlinear parabolic equations,”Res. Repts. Inst. Inform. Sci. Technol.,6, 45–57 (1980).Google Scholar
  9. 9.
    A. S. Kalashnikov, “The Cauchy problem for degenerate second-order parabolic equations with non-power non-linearities,”Trudy Sem. Petrovsk,6, 83–96 (1981).MATHMathSciNetGoogle Scholar
  10. 10.
    H. W. Alt and S. Luckhaus, “Quasi-linear elliptic-parabolic differential equations,”Math. Z.,183, 311–341 (1983).CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    M. Tsutsumi, “On solutions of some doubly nonlinear degenerate parabolic equations with absorption,”J. Math. Anal. Appl.,132, 187–212 (1988).CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    D. Blanchard and G. Francfort, “Study of a doubly nonlinear heat equation with no growth assumption on the parabolic term,”SIAM J. Math. Anal.,19, 1032–1056 (1988).CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Y. Jingxue, “On a class of quasilinear parabolic equations of second order with double degeneracy,”J. Partial Diff. Eq.,3, 49–64 (1990).MATHGoogle Scholar
  14. 14.
    G. I. Barenblatt, “On some unsteady motions of a fluid and a gas in a porous medium,”Prikl. Mat. Mekh.,16, 67–78 (1952).MATHMathSciNetGoogle Scholar
  15. 15.
    O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva,Linear and Quasilinear Equations of Parabolic Type [in Russian], Moscow (1967).Google Scholar
  16. 16.
    O. A. Oleinik and S. N. Kruzhkov, “Qusilinear second order parabolic equations with several independent variables,”Usp. Mat. Nauk,16, No. 5(101), 115–155 (1961).Google Scholar
  17. 17.
    J. G. Aronson and J. Serrin, “Local behavior of solutions of quasilinear parabolic equations,”Arch. Rational. Mech. Anal.,25, 81–122 (1967).MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    A. V. Ivanov, “Harnack's inequality for generalized solutions of parabolic equations of second order,”Trudy Mat. Inst. Steklov,102, 51–84 (1967).MATHMathSciNetGoogle Scholar
  19. 19.
    N. S. Trudinger, “Pointwise estimates and quasi-linear parabolic equations,”Commun. Pure Appl. Math.,21, 205–226 (1968).MATHMathSciNetGoogle Scholar
  20. 20.
    L. A. Caffarelli and A. Friedman, “Regularity of the free-boundary of a gas in a n-dimensional porous medium,”Indiana Univ. Math. J.,29, 361–369 (1980).CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    L. A. Caffarelli and L. C. Evans, “Continuity of the temperature in the two phase Stefan problem,”Arch. Rat. Mech. Anal.,81, 199–220 (1983).CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    P. E. Sacks, “Continuity of solutions of a singular parabolic equation,”Nonl. Anal. TMA,7, 387–409 (1983).CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    E. DiBenedetto, “Continuity of weak solutions to a general porous medium equation,”Indiana Univ. Math. J.,32, No. 1, 83–118 (1983).CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    W. Ziemer, “Interior and boundary continuity of weak solutions of degenerate parabolic equations,”Trans. Am. Math. Soc.,271, 733–748 (1982).CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    A. V. Ivanov, “On estimate of modulus of continuity of weak solutions of some singular parabolic equations,”Zap. Nauchn. Semin. LOMI,147, 49–72 (1985).MATHGoogle Scholar
  26. 26.
    Chen Ya-Zhe, “Hölder estimates for solutions of uniformly degenerate parabolic equations,”Chin. Ann. Math.,5b(4), 666–678 (1984).Google Scholar
  27. 27.
    E. DiBenedetto and A. Friedman, “Hölder estimates for nonlinear degenerate parabolic systems,”J. Reine Angew. Math.,357, 83–127 (1985).MathSciNetGoogle Scholar
  28. 28.
    A. V. Ivanov, “Estimates of the Hölder constant for weak solutions of degenerate parabolic equations,”Zap. Nauchn. Sem. LOMI,152, 21–44 (1986).MATHGoogle Scholar
  29. 29.
    M. Wiegner, “OnC α-regularity of the gradient of solutions of degenerate parabolic system,” Preprint, Universitat Bonn, No. 682 (1984).Google Scholar
  30. 30.
    E. DiBenedetto, “On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients,”Ann. Sc. Norm. Sur.,13, No. 3, 485–535 (1986).MathSciNetGoogle Scholar
  31. 31.
    Chen Ya-Zhe and E. DiBenedetto, “On the local behaviour of solutions of singular parabolic equations,”Arch. Rat. Mech. Anal.,103, No. 4, 319–346 (1983).Google Scholar
  32. 32.
    A. V. Ivanov, “Hölder estimates for weak solutions of quasilinear doubly degenerate parabolic equations,”Zap. Nauchn. Semin. LOMI,171, 70–105 (1989).MATHGoogle Scholar
  33. 33.
    A. V. Ivanov, “Uniform Hölder estimates for weak solutions of quasilinear doubly degenerate parabolic equations,”Algebra Analiz,3, No. 2, 139–179 (1991).MATHGoogle Scholar
  34. 34.
    A. V. Ivanov, “Boundary Hölder estimates for weak solutions of quasilinear doubly degenerate parabolic equations,”Zap. Nauchn. Semin. LOMI,183, 45–69 (1991).Google Scholar
  35. 35.
    A. V. Ivanov, and P. Z. Mkrtchan, “On the existence Hölder continuous weak solutions of the first boundary value problem for quasilinear doubly degenerate parabolic equations,”Zap. Nauchn. Semin. LOMI,182, 5–28 (1990).MATHGoogle Scholar
  36. 36.
    A. V. Ivanov and P. Z. Mkrtchan, “On the regularity up to the boundary of weak solutions of the Cauchy-Dirichlet problem for quasilinear doubly degenerate parabolic equations,”Zap. Nauchn. Semin. LOMI,191, 83–98 (1991).Google Scholar
  37. 37.
    A. V. Ivanov, “The classesB m,l and Hölder estimates of weak solutions for quasilinear doubly degenerate parabolic equations,” Preprints POMI (1991) E-11-91, 3-66; E-12-91, 3-51.Google Scholar
  38. 38.
    A. V. Ivanov, “The classesB m,l and Hölder estimates of weak solutions for quasilinear parabolic equations admitting doubly degeneracy”Zap. Nauchn. Semin. POMI,197, 42–70 (1992).MATHGoogle Scholar
  39. 39.
    J. R. Esteban and J. L. Vazques, “On the equation of turbulent filtration in one-dimentional porous media,”Nonl. Anal. TMA,10, No. 11, 1303–1325 (1989).CrossRefGoogle Scholar
  40. 40.
    M. M. Porzio and V. Vespri, “Hölder estimates for local solutions on some doubly nonlinear parabolic equations,” Preprint (1992).Google Scholar
  41. 41.
    V. Vespri, “On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations,”Manuscripta Math.,75, 65–80 (1992).MATHMathSciNetGoogle Scholar
  42. 42.
    V. Vespri, “Harnack type inequalities for solutions of certain doubly nonlinear parabolic equations,”Universita degli studi di Milano,32, 1–23 (1991).MathSciNetGoogle Scholar
  43. 43.
    A. V. Ivanov, “Quasilinear doubly degenerate and singular-degenerate equations,”to appear in “Vortragsreihe 1992”, Mathematishes Institut der Bonn Universitat.Google Scholar
  44. 44.
    A. V. Ivanov, “Equations of the type of slow and normal diffusion,”to appear in Zap. Nauch. Semin. POMI.Google Scholar
  45. 45.
    A. V. Ivanov, “Equations of the type of fast diffusion,”to appear in St. Petersburg Math. J. Google Scholar
  46. 46.
    B. H. Gilding and L. A. Peletier, “The Cauchy Problem for an equation in the theory of infiltration,”Arch. Rat. Mech. Anal.,61, No. 2, 127–140 (1976).MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    O. A. Ladyzhenskaya, “On construction of discontinuous solutions of quasilinear hyperbolic equations as limits of solutions of correspondent parabolic equations under tending to zero of the viscosity coefficients,”Trudy Moskovsk. Matem. Obsch.,6, 465–480 (1957).MATHGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • A. V. Ivanov

There are no affiliations available

Personalised recommendations