Citation: |
[1] |
L. Ansini and L. Giacomelli, Doubly nonlinear thin-film equations in one space dimension, Arch. Rational Mech. Anal., 173 (2004), 89-131.doi: 10.1007/s00205-004-0313-x. |
[2] |
F. Bernis, Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equations, 1 (1996), 337-368. |
[3] |
F. Bernis, Finite speed of propagation for thin viscous flows when $ 2 \le n < 3 $, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 1169-1174. |
[4] |
F. Bernis and A. Friedman, Higher-order nonlinear degenerate parabolic equations, J. Diff. Equations, 83 (1990), 179-206.doi: 10.1016/0022-0396(90)90074-Y. |
[5] |
F. Bernis, L. A. Peletier and S. M. Williams, Source type solutions of a fourth order nonlinear degenerate parabolic equation, Nonlinear Anal., 18 (1992), 217-234.doi: 10.1016/0362-546X(92)90060-R. |
[6] |
A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: the moving contact line with a porous media cutoff of the van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564. |
[7] |
A. L. Bertozzi and M. C. Pugh, The lubrication approximation for viscous films: regularity and long time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123.doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.3.CO;2-V. |
[8] |
A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661.doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.3.CO;2-2. |
[9] |
M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv Differ. Equ., 3 (1998), 417-440. |
[10] |
E. Bertta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Ration. Mech. Anal., 129 (1995), 175-200. |
[11] |
M. Boutat, S. Hilout, J. E. Rakotoson and J. M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Analysis, 69 (2008), 1268-1286.doi: 10.1016/j.na.2007.06.028. |
[12] |
M. Boutat, S. Hilout, J. E. Rakotoson, J. M. Rakotoson, The generalized thin film equation with periodic-domain conditions, Applied Mathematics Letters, 21 (2008), 101-104.doi: 10.1016/j.aml.2007.02.014. |
[13] |
E. A. Carlen and S. Ulusory, An entropy dissipation-entropy estimate for a thin film type equation, Comm. Math. Sci., 3 (2005), 171-178. |
[14] |
J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.doi: 10.1007/s006050170032. |
[15] |
J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin film equation, Commun. Math. Phys., 225 (2002), 551-571.doi: 10.1007/s002200100591. |
[16] |
M. Chugunova, M. Pugh and R. Taranets, Research announcement: Finite-time blow up and long-wave unstable thin film equations, preprint, arXiv:1008.0385. |
[17] |
R. Dal Passo, H. Garcke and G. Grün, On a fourth order degenerate parabolic equation: global entropy estimates, existence, and qualitative behaviour of solutions, SIAM J. Math. Anal., 29 (1998), 321-342. |
[18] |
R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomenon for thin film equations, Ann. Scuola Norm. Sup. Pisa, 30 (2001), 437-463. |
[19] |
R. Dal Passo, L. Giacomelli and A. Shishkov, The thin film equation with nonlinear diffusion, Comm. Partial Differential Equations, 26 (2001), 1509-1557. |
[20] |
S. D. Èĭdel'man, Parabolic Systems, Translated from the Russian by Scripta Technica, London, North-Holland Publishing Co., Amsterdam, 1969. |
[21] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London,1969. |
[22] |
L. Giacomelli, A fourth-order degenerate parabolic equation describing thin viscous flows over an inclined plane, Applied Mathematics Letters, 12 (1999), 107-111.doi: 10.1016/S0893-9659(99)00130-5. |
[23] |
L. Giacomelli and A. Shishkov, Propagation of support in one-dimensional convected thin-film flow, Indiana Univ. Math. J., 54 (2005), 1181-1215.doi: 10.1512/iumj.2005.54.2532. |
[24] |
G. Grün, Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal harding, Z. Anal. Anwendungen., 14 (1995), 541-574. |
[25] |
G. Grün, "On Free Boundary Problems Arising in Thin Film Flow," Habilitation thesis, University of Bonn, 2001. |
[26] |
G. Grün, Droplet spreading under weak slippage: the waiting time phenomenon, Ann. I. H. Poincaré–AN, 21 (2004), 255-269.doi: 10.1016/j.ahihpc.2003.02.002. |
[27] |
L. M. Hocking, Spreading and instability of a viscous fluid sheet, Journal of Fluid Mechanics, 211 (1990), 373-392.doi: 10.1017/S0022112090001616. |
[28] |
J. R. King, Two generalisations of the thin film equation, Math. Comput. Modelling, 34 (2001), 737-756.doi: 10.1016/S0895-7177(01)00095-4. |
[29] |
J. J. Li, On a fourth order degenerate parabolic equation in higher space dimensions, Journal of Mathematical Physics, 50 (2009), 123524, 26 pp. Available from: http://dx.doi.org/10.1063/1.3272788.doi: 10.1063/1.3272788. |
[30] |
X. Liu and C. Qu, Finite speed of propagation for thin viscous flows over an inclined plane, Nonlinear Anal. Real World Appl., 13 (2012), 464-475.doi: 10.1016/j.nonrwa.2011.08.003. |
[31] |
E. Momoniata, T. G. Myers and S. Abelman, Similarity solutions of thin film flow driven by gravity and surface shear, Nonlinear Analysis: Real World Applications, 10 (2009), 3443-3450.doi: 10.1016/j.nonrwa.2008.10.070. |
[32] |
A. Oron, S. H, Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Modern Phys., 69 (1997), 931-980.doi: 10.1103/RevModPhys.69.931. |
[33] |
E. O. Tuck and L. W. Schwaxtz, Thin static drops with a free attachment boundary, Journal of Fluid Mechanics, 223 (1991), 313-324. |
[34] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics" (Vol. I), Amsterdam: North-Holland, (2002), 71-305. |