Journal d'Analyse Mathématique

, Volume 137, Issue 1, pp 1–55 | Cite as

Global estimates for generalized Forchheimer flows of slightly compressible fluids

  • Luan HoangEmail author
  • Thinh Kieu


This paper is focused on the generalized Forchheimer flows of slightly compressible fluids in porous media. They are reformulated as a degenerate parabolic equation for the pressure. The initial boundary value problem is studied with time-dependent Dirichlet boundary data. The estimates up to the boundary and for all time are derived for the L-norm of the pressure, its gradient and time derivative. Large-time estimates are established to be independent of the initial data. Particularly, thanks to the special structure of the pressure’s nonlinear equation, the global gradient estimates are obtained in a relatively simple way, avoiding complicated calculations and a prior requirement of Hölder estimates.


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  1. [1]
    E. Aulisa, L. Bloshanskaya, L. Hoang, and A. Ibragimov, Analysis of generalized Forchheimer flows of compressible fluids in porous media, J. Math. Phys. 50 103102 (2009).Google Scholar
  2. [2]
    J. Bear, Dynamics of Fluids in Porous Media, Dover Publications, 1988. Reprint of the American Elsevier Publishing Company, Inc., New York, 1972 edition.zbMATHGoogle Scholar
  3. [3]
    H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, 1856.Google Scholar
  4. [4]
    E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.CrossRefzbMATHGoogle Scholar
  5. [5]
    L. Dung, Ultimately uniform boundedness of solutions and gradients for degenerate parabolic systems, Nonlinear Anal. 39(2000), 157–171.Google Scholar
  6. [6]
    J. Dupuit, Mouvement de l’eau a travers le terrains permeables, C. R. Hebd. Seances Acad. Sci. 45 (1857), 92–96.Google Scholar
  7. [7]
    P. Forchheimer, Wasserbewegung durch Boden, Zeit. Ver. Deut. Ing. 45 (1901), 1781–1788.Google Scholar
  8. [8]
    P. Forchheimer, Hydraulik, 3rd edition, Leipzig, Berlin, B. G. Teubner. 1930.zbMATHGoogle Scholar
  9. [9]
    L. Hoang and A. Ibragimov, Structural stability of generalized Forchheimer equations for compressible fluids in porous media, Nonlinearity 24 (2011), 1–41.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    L. Hoang and A. Ibragimov, Qualitative study of generalized Forchheimer flows with the flux boundary condition, Adv. Differential Equations 17 (2012), 511–556.MathSciNetzbMATHGoogle Scholar
  11. [11]
    L. Hoang, A. Ibragimov, T. Kieu, and Z. Sobol, Stability of solutions to generalized Forchheimer equations of any degree, J. Math. Sci. 210 (2015), 476–544.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    L. Hoang and T. Kieu, Interior estimates for generalized Forchheimer flows of slightly compressible fluids, Adv. Nonlinear Stud. 17 (2017), 739–768.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    L. T. Hoang, T. T. Kieu, and T. V. Phan, Properties of generalized Forchheimer flows in porous media, J. Math. Sci. 202 (2014), 259–332.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.CrossRefGoogle Scholar
  15. [15]
    G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996.CrossRefzbMATHGoogle Scholar
  16. [16]
    M. Muskat, The Flow of Homogeneous Fluids through Porous Media, McGraw-Hill Book Company, Inc., 1937.zbMATHGoogle Scholar
  17. [17]
    D. A. Nield and A. Bejan, Convection in Porous Media, fourth edition, Springer-Verlag, New York, 2013.CrossRefzbMATHGoogle Scholar
  18. [18]
    F. Ragnedda, S. Vernier Piro, and V. Vespri, Asymptotic time behaviour for non-autonomous degenerate parabolic problems with forcing term, Nonlinear Anal. 71 (2009), e2316–e2321.Google Scholar
  19. [19]
    F. Ragnedda, S. Vernier Piro, and V. Vespri, Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations, Math. Ann. 348 (2010), 779–795.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.CrossRefzbMATHGoogle Scholar
  21. [21]
    B. Straughan, Stability and Wave Motion in Porous Media, Springer, New York, 2008.zbMATHGoogle Scholar
  22. [22]
    M. D. Surnachëv, On improved estimates for parabolic equations with double degeneration, Proc. Steklov Inst. Math. 278 (2012), 241–250.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, second edition, Springer-Verlag, New York, 1997.CrossRefzbMATHGoogle Scholar
  24. [24]
    J. L. Vázquez, Smoothing and decay estimates for nonlinear diffusion equations, Oxford University Press, Oxford, 2006.CrossRefzbMATHGoogle Scholar
  25. [25]
    J. L. Vázquez, The Porous Medium Equation, The Clarendon Press Oxford University Press, Oxford, 2007.zbMATHGoogle Scholar
  26. [26]
    J. C. Ward. Turbulent flow in porous media, J. Hydraulics Division 90 (1964), 1–12.Google Scholar

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsTexas Tech UniversityLubbockUSA
  2. 2.Department Of MathematicsUniversity Of North Georgia, Gainesville CampusOakwoodUSA

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