A waiting time phenomenon for thin film equations

Roberta Dal Passo; Lorenzo Giacomelli; Günther Grün

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2001)

  • Volume: 30, Issue: 2, page 437-463
  • ISSN: 0391-173X

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Dal Passo, Roberta, Giacomelli, Lorenzo, and Grün, Günther. "A waiting time phenomenon for thin film equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 30.2 (2001): 437-463. <http://eudml.org/doc/84448>.

@article{DalPasso2001,
author = {Dal Passo, Roberta, Giacomelli, Lorenzo, Grün, Günther},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {fourth-order degenerate parabolic equation; weighted energy estimates; Neumann boundary conditions; Gagliardo-Nirenberg inequalities; entropy},
language = {eng},
number = {2},
pages = {437-463},
publisher = {Scuola normale superiore},
title = {A waiting time phenomenon for thin film equations},
url = {http://eudml.org/doc/84448},
volume = {30},
year = {2001},
}

TY - JOUR
AU - Dal Passo, Roberta
AU - Giacomelli, Lorenzo
AU - Grün, Günther
TI - A waiting time phenomenon for thin film equations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2001
PB - Scuola normale superiore
VL - 30
IS - 2
SP - 437
EP - 463
LA - eng
KW - fourth-order degenerate parabolic equation; weighted energy estimates; Neumann boundary conditions; Gagliardo-Nirenberg inequalities; entropy
UR - http://eudml.org/doc/84448
ER -

References

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  1. [ 1 ] N.D. Alikakos, On the pointwise behavior of the solutions of the porous medium equation as t approaches zero or infinity, Nonlinear Anal.9 (1985), 1095-1113. Zbl0589.35064MR806912
  2. [2] D.G. Aronson, "The porous medium equation", In A. Dold and B. Eckmann, editors, Nonlinear Diffusion Problems. Lecture Notes in Mathematics, 1224, Springer-Verlag, 1985. Zbl0626.76097MR877986
  3. [3] E. Beretta - M. Bertsch - R. Dal Passo, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation, Arch. Rat. Mech. Anal.129 (1995), 175-200. Zbl0827.35065MR1328475
  4. [4] F. Bernis, Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems, In:"Free boundary problems: theory and applications", J. I. Diaz - M. A. Herrero - A. Linan - J. L. Vazquez (eds.), Pitman Research Notes in Mathematics323, Longman, Harlow, 1995, pp. 40-56. Zbl0839.35102MR1342325
  5. [5] F. Bernis, Finite speed ofpropagation and continuity of the interfacefor thin viscous flows, Adv. Differential Equations1 no. 3 (1996), 337-368. Zbl0846.35058MR1401398
  6. [6] F. Bernis, Finite speed of propagation for thin viscous flows when 2 ≤ n &lt; 3, C.R. Acad. Sci. Paris Sér. I Math.322 (1996). Zbl0853.76018
  7. [7] F. Bernis - A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations83 (1990), 179-206. Zbl0702.35143MR1031383
  8. [8] F. Bernis - L.A. Peletier - S.M. Williams, Source-type solutions of a fourth order nonlinear degenerate parabolic equations, Nonlinear Anal.18 (1992), 217-234. Zbl0778.35056MR1148286
  9. [9] A. Bertozzi - M. Pugh, The lubrication approximation for thin viscous films: the moving contact line with a porous media cut off of van der waals interactions, Nonlinearity7 (1994), 1535-1564. Zbl0811.35045MR1304438
  10. [10] A.L. Bertozzi - M. Pugh, The lubrication approximation for thin viscous films: regularity and long time behaviour of weak solutions, Nonlinear Anal.18 (1992), 217-234. 
  11. [11] M. Bertsch - R. Dal Passo - H. Garcke - G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations3 (1998), 417-440. Zbl0954.35035MR1751951
  12. [12] R. Dal Passo - H. Garcke, Solutions of a fourth order degenerate parabolic equation with weak initial trace, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 153-181. Zbl0945.35049MR1679081
  13. [13] R. Dal Passo - H. Garcke - G. Grün, On a fourth order degenerate parabolic equation: global entropy estimates and qualitative behaviour of solutions, SIAM J. Math. Anal.29 (1998), 321-342. Zbl0929.35061MR1616558
  14. [14] R. Dal Passo - L. Giacomelli - A. Shishkov, The thin film equation with nonlinear diffusion, Preprint Me.Mo.Mat. Department 2/2000, to appear in Comm. Partial Differential Equations. Zbl1001.35070MR1865938
  15. [15] E.B. Dussan - S. Davis, On the motion of a fluid-fluid interface along a solid surface, J. Fluid Mech.65 (1974), 71-95. Zbl0282.76004
  16. [16] R. Ferreira - F. Bernis, Source-type solutions to thin-film equations in higher space dimensions, European J. Appl. Math.8 (1997), 507-524. Zbl0894.76019MR1479525
  17. [17] E. Gagliardo, Ulteriori properità di alcune classi di funzioni in piú variabili, Ricerche di Mat. (1959), 24-51. Zbl0199.44701MR109295
  18. [18] G. Grün, Degenerate parabolic equations of fourth order and a plasticity model with nonlocal hardening, Z. Anal. Anwendungen14 (1995), 541-573. Zbl0835.35061MR1362530
  19. [19] G. Grün - M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Numer. Mathematik87 (2000), 113-152. Zbl0988.76056MR1800156
  20. [20] G. Grün - M. Rumpf, Entropy consistent finite volume schemes for the thin film equation, In: "Finite volume schemes for complex applications II", R. Vilsmeier - F. Benkhaldoun - D. Hänel (eds.), Hermes Science Publications, Paris, 1999, pp. 205-214. Zbl1052.65526MR2062140
  21. [21] J. Hulshof - A. Shishkov, The thin film equation with 2 ≤ n &lt; 3: finite speed of propagation in terms of the l1-norm, Adv. Differential Equations3 (1998), 625-642. Zbl0953.35072
  22. [22] B.F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc.234 (1977), 381-415. Zbl0365.35030MR492856
  23. [23] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, Cl. Sci.13 (1959), 115-162. Zbl0088.07601MR109940
  24. [24] A. Oron - S.H. Davis - S.G. Bankoff, Long-scale evolution of thin liquid films, Reviews of Modem Physics69 (1997), 932-977. 
  25. [25] F. Otto, Lubrication approximation with prescribed non-zero contact angle: an existence result, Comm. Partial Differential Equations23 (1998), 2077-2164. Zbl0923.35211MR1662172
  26. [26] N.F. Smyth - J.M. Hill, Higher order nonlinear diffusion, IMA J. Applied Mathematics40 (1988), 73-86. Zbl0694.35091MR983990
  27. [27] G. Stampacchia, "Équations elliptiques du second ordre à coefficients discontinus", Les presses de l'université de Montréal, Montréal, 1966. Zbl0151.15501MR251373
  28. [28] J.L. Vazquez, An introduction to the mathematical theory of the porous medium equation, In: "Shape Optimization and Free Boundaries", M. C. Delfour-G. Sabidussi (eds.), Kluwer Academic Publishers, Netherlands, 1992, pp. 347-389. Zbl0765.76086MR1260981

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