Advertisement

Boundedness of Solutions to a Degenerate Diffusion Equation

  • Pavel KrejčíEmail author
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 22)

Abstract

The diffusion equation with a bounded saturation range under the time derivative and with Robin boundary conditions is shown to admit a regular bounded solution provided that the saturation function and the permeability coefficient have controlled decay at infinity. The result remains valid even if Preisach hysteresis is present in the pressure-saturation relation. The method of proof is based on a Moser-Alikakos iteration scheme which is compatible with a generalized Preisach energy dissipation mechanism.

Keywords

Degenerate parabolic equation Diffusion Hysteresis 

AMS (MOS) Subject Classification

35K65 47J40 

References

  1. 1.
    Albers, B.: Modeling the hysteretic behavior of the capillary pressure in partially saturated porous media – a review. Acta Mech. 225, 2163–2189 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Albers, B., Krejčí, P.: Unsaturated porous media flow with thermomechanical interaction. Math. Methods Appl. Sci. 39, 2220–2238 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Alikakos, N.D.: An application of the invariance principle to reaction-diffusion equations. J. Differ. Equ. 33, 201–225 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z 183, 311–341 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Alt, H.W., Luckhaus, S., Visintin, A.: On nonstationary flow through porous media. Ann. Mat. Pura Appl. IV. Ser. 136, 303–316 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Aronson, D.G., Bénilan, P.: Régularité des solutions de l’équation des milieux poreux dans \(\mathbb{R}^{N}\). C. R. Acad. Sci. Paris, Sér. A 288, 103–105 (1979)zbMATHGoogle Scholar
  7. 7.
    Bagagiolo, F., Visintin, A.: Hysteresis in filtration through porous media. Z. Anal. Anwendungen 19, 977–997 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bagagiolo, F., Visintin, A.: Porous media filtration with hysteresis. Adv. Math. Sci. Appl. 14, 379–403 (2004)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Besov, O.V., Il’in, V.P., Nikol’skiĭ, S.M.: Integral Representations of Functions and Imbedding Theorems. Scripta Series in Mathematics. Halsted Press/Wiley, New York,Toronto, London (vol. I, 1978), (vol. II, 1979) [Russian version Nauka, Moscow (1975)]Google Scholar
  10. 10.
    Brokate, M., Visintin, A.: Properties of the Preisach model for hysteresis. J. Reine Angew. Math. 402, 1–40 (1989)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Colli, P., Gilardi, G., Podio-Guidugli, P., Sprekels, J.: Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity. J. Differ. Equ. 254, 4217–4244 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Detmann, B., Krejčí, P., Rocca, E.: Solvability of an unsaturated porous media flow problem with thermomechanical interaction. SIAM J. Math. Anal. 48, 4175–4201 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Droniou, J., Eymard, R., Talbot, K.S.: Convergence in C([0, T]; L 2(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations. J. Differ. Equ. 260, 7821–7860 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Flynn, D., McNamara, H., O’Kane. J.P., Pokrovskiĭ, A.V.: Application of the Preisach model to soil-moisture hysteresis. In: Bertotti, G., Mayergoyz, I. (eds.) The Science of Hysteresis, vol. 3, pp. 689–744. Academic, Oxford (2006)Google Scholar
  15. 15.
    Gilardi, G.: A new approach to evolution free boundary problems. Commun. Partial Differ. Equ. 4, 1099–1122 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Haverkamp, R., Reggiani, P., Ross, P.J., Parlange, J.-Y.: Soil water hysteresis prediction model based on theory and geometric scaling. In: Raats, P.A.C., Smiles, D., Warrick, A.W. (eds.) Environmental Mechanics, Water, Mass and Energy Transfer in the Biosphere, pp. 213–246. American Geophysical Union, Washington (2002)CrossRefGoogle Scholar
  17. 17.
    Hilpert, M.: On uniqueness for evolution problems with hysteresis. In: Rodrigues, J.F. (ed.) Mathematical Models for Phase Change Problems, pp. 377–388. Birkhäuser, Basel (1989)CrossRefGoogle Scholar
  18. 18.
    Krasnosel’skiĭ, M.A., Pokrovskiĭ, A.V.: Systems with Hysteresis. Springer, Berlin (1989). Nauka, Moscow (1983) [Russian edn.]Google Scholar
  19. 19.
    Krejčí, P.: On Maxwell equations with the Preisach hysteresis operator: the one-dimensional time-periodic case. Apl. Mat. 34, 364–374 (1989)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Krejčí, P.: Hysteresis memory preserving operators. Appl. Math. 36, 305–326 (1991)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Krejčí, P.: Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gakuto International Series: Mathematical Sciences and Applications, vol. 8. Gakkotōsho, Tokyo (1996)Google Scholar
  22. 22.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Science, Providence, RI (1968)zbMATHGoogle Scholar
  23. 23.
    Preisach, F.: Über die magnetische Nachwirkung. Z. Phys. 94, 277–302 (1935)CrossRefGoogle Scholar
  24. 24.
    Showalter, R.E., Stefanelli, U.: Diffusion in poro-plastic media. Math. Methods Appl. Sci. 27, 2131–2151 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Tao, Y., Winkler, M.: Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity. J. Differ. Equ. 252, 692–715 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of MathematicsAcademy of Sciences of the Czech RepublicPraha 1Czech Republic

Personalised recommendations