Abstract.
We consider a free-boundary problem for a class of fourth-order nonlinear parabolic equations which are degenerate both with respect to the unknown and to its third derivative. The problem is relevant in the description of the surface-tension driven spreading of a non-Newtonian liquid over a solid surface in the “complete wetting” regime. Relying solely on global and local energy estimates and on Bernis’ inequalities, we prove existence of solutions to this problem, and obtain sharp upper bounds for the propagation of their support. A necessary condition for the occurrence of waiting-time phenomena is also derived.
Similar content being viewed by others
References
Almgren, R.: Singularity formation in Hele-Shaw bubbles. Phys. Fluids 8, 344–352 (1996)
Andreucci, D., Tedeev, A.F.: Finite speed of propagation for the thin-film equation and other higher-order parabolic equations with general nonlinearity. Interfaces Free Bound. 3, 233–264 (2001)
Ansini, L., Giacomelli, L.: Shear-thinning liquid films: Macroscopic and asymptotic behavior by quasi self-similar solutions. Nonlinearity 15, 2147–2164 (2002)
Beretta, E., Bertsch, M., Dal Passo, R.: Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. Arch. Rational Mech. Anal. 129, 175–200 (1995)
Bernis, F.: Existence results for doubly nonlinear higher order parabolic equations on unbounded domains. Math. Ann. 279, 373–394 (1988)
Bernis, F., Friedman, A.: Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83, 179–206 (1990)
Bernis, F.: Integral inequalities with applications to nonlinear degenerate parabolic equations. In: Angell, T.S., Cook, L.P., Kleinman, R.E. Olmstead, W.E. Nonlinear Boundary Value Problems. SIAM, Philadelphia, 1996
Bernis, F.: Finite speed of propagation and continuity of the interface for thin viscous flows. Adv. Differential Equations 1, 337–368 (1996)
Bernis, F.: Finite speed of propagation for thin viscous flows when 2 n≤3. C.R. Acad. Sci. Paris, Sér. I Math. 322, 1169–1174 (1996)
Bertozzi, A.L., Pugh, M.: The lubrication approximation for thin viscous films: The moving contact line with a ‘‘porous media’’ cut-off of van der Waals interactions. Nonlinearity 7, 1535–1564 (1994)
Bertozzi, A.L., Pugh, M.: The lubrication approximation for thin viscous films: regularity and long time behavior of weak solutions. Comm. Pure Appl. Math. 49, 85–123 (1996)
Bertozzi, A.L., Pugh, M.: Long-wave instabilities and saturation in thin film equations. Comm. Pure Appl. Math. 51, 625–661 (1998)
Bertsch, M., Dal Passo, R., Garcke, H., Grün, G.: The thin viscous flow equation in higher space dimension. Adv. Differential Equations 3, 417–440 (1998)
Betelú, S.I., Fontelos, M.A.: Capillarity driven spreading of power-law fluids. Appl. Math. Lett. 16, 1315–1320 (2003)
Bird, R.B., Armstrong, R.C., Hassager, O.: Dynamics of polymeric liquids. John Wiley and Sons, 1977
Carré, A., Eustache, F.: Spreading kinetics of shear-thinning fluids in wetting and dewetting models. Langmuir 16, 2936–2941 (2000)
Dal Passo, R., Garcke, H.: Solutions of a fourth-order degenerate parabolic equation with weak initial trace Ann. Sc. Norm. Sup. Pisa Cl. Sci. 28, 153–181 (1999)
Dal Passo, R., Garcke, H., Grün, G.: On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions. SIAM J. Math. Anal. 29, 321–342 (1998)
Dal Passo, R., Giacomelli, L., Grün, G.: A waiting time phenomenon for thin film equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30, 437–463 (2001)
Dal Passo, R., Giacomelli, L., Grün, G.: Waiting time phenomena for degenerate parabolic equations – a unifying approach. In Geometric Analysis and Nonlinear Partial Differential Equations, S. Hildebrandt & H. Karcher, Eds., Springer-Verlag, Berlin Heidelberg New York, 2003, pp. 637–648
Dal Passo, R., Giacomelli, L., Shishkov, A.E.: The thin film equation with nonlinear diffusion. Comm. Partial Differential Equations 26, 1509–1557 (2001)
de Gennes, P.G.: Wetting: Statics and dynamics. Rev. Mod. Phys. 57, 827–863 (1985)
DiBenedetto, E.: Degenerate parabolic equations. Springer-Verlag, New York, 1993
Dussan V., E.B., Davis, S.H.: On the motion of fluid-fluid interface along a solid surface. J. Fl. Mech. 65, 71–95 (1974)
Gagliardo, E.: Ulteriori proprietà di alcune classi di funzioni in piú variabili. Ricerche di Mat. Napoli, 8, 24–51 (1959)
Giacomelli, L.: A fourth order degenerate parabolic equation describing thin viscous flows over an inclined plane. Appl. Math. Lett. 12, 107–111 (1999)
Giacomelli, L., Otto, F.: Variational formulation for the lubrication approximation of the Hele-Shaw flow. Calc. Var. Partial Differential Equations 13, 377–403 (2001)
Giacomelli, L., Otto, F.: Droplet spreading: Intermediate scaling law by PDE methods. Comm. Pure Appl. Math. 55, 217–254 (2002)
Giacomelli, L., Otto, F.: Rigorous lubrication approximation. Interfaces Free Bound. 5, 483–529 (2003)
Grün, G.: Degenerate parabolic differential equations of fourth order and a plasticity model with non-local hardening. Z. Anal. Anwendungen 14, 541–574 (1995)
Grün, G.: On free boundary problems arising in thin film flow. Habilitation thesis, University of Bonn, 2001
Grün, G.: On Bernis’ interpolation inequalities in multiple space dimensions. Z. Anal. Anwendungen 20, 987–998 (2001)
Grün, G.: Droplet spreading under weak slippage: the optimal asymptotic propagation rate in the multi-dimensional case. Interfaces Free Bound. 4, 309–323 (2002)
Grün, G.: Droplet spreading under weak slippage: the waiting time phenomenon. Ann. I.H.P., Analyse non lineaire, To appear.
Huh, C., Scriven, L.E.: Hydrodynamic model of a steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85–101 (1971)
Hulshof, J., Shishkov, A.E.: The thin film equation with 2 n ≤ 3: finite speed of propagation in terms of the L1-norm. Adv. Differential Equations 3, 625–642 (1998)
King, J.R.: Two generalisations of the thin film equation. Mathematical and Computer Modelling 34, 737–756 (2001)
King, J.R.: Thin-film flows and higher-order degenerate parabolic equations. IUTAM Symposium on Free Surface Flows, A.C. King & Y.D. Shikhmurzaev (eds), Kluwer Academic Publishers, 2001, pp. 7–18
King, J.R.: The spreading of power-law fluids. IUTAM Symposium on Free Surface Flows, A.C. King & Y.D. Shikhmurzaev (eds.), Kluwer Academic Publishers, 2001, pp. 153–160
Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13, 115–162 (1959)
Opic, B., Kufner, A.: Hardy-type inequalities. Pitman Res. Notes 219, Longman, 1990
Oron, A., Davis, S.H., Bankoff, G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980 (1997)
Otto, F.: Lubrication approximation with prescribed non-zero contact angle. Comm. Partial Differential Equations 23, 2077–2164 (1998)
Shishkov, A.E.: Estimates of rate of propagation of perturbations in quasilinear divergent degenerate parabolic equations of high order. Ukrain. Math. J. 44, 1335–1340 (1992)
Shishkov, A.E., Shchelkov, A.G.: Dynamics of the supports of energy solutions of mixed problems for quasilinear parabolic equations of arbitrary order. Izv. Math. 62, 601–626 (1998)
Simon, J.: Compact Sets in the Space Lp(0,T; B). Annali Mat. Pura Appl. 146, 65–97 (1987)
Stampacchia, G.: Équations elliptiques du second ordre à coefficients discontinus. Les presses de l‘Université de Montréal, Montréal, 1966
Weidner, D.E., Schwartz, L.W.: Contact-line motion of shear-thinning liquids. Phys. Fluids 6, 3535–3538 (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Otto
Rights and permissions
About this article
Cite this article
Ansini, L., Giacomelli, L. Doubly Nonlinear Thin-Film Equations in One Space Dimension. Arch. Rational Mech. Anal. 173, 89–131 (2004). https://doi.org/10.1007/s00205-004-0313-x
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00205-004-0313-x