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March  2014, 13(2): 543-566. doi: 10.3934/cpaa.2014.13.543

Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation

Lihua Min 1, and  Xiaoping Yang 1,

1. 

School of Science, Nanjing University of Science and Technology, Nanjing, 210094, China, China

Received  October 2012 Revised  July 2013 Published  October 2013

We consider a fourth order degenerate equation describing thin films over an inclined plane in this paper. A new approximating problem is introduced in order to obtain the local energy estimate of the solution. Based on combined use of local entropy estimate, local energy estimate and the suitable extensions of Stampacchia's Lemma to systems, we obtain the finite speed of propagation property of strong solutions, which has been known for the case of strong slippage $ n<2, $ in the case of weak slippage $ 2 \leq n < 3. $ The long time behavior of positive classical solutions is also discussed. We apply the entropy dissipation method to quantify the explicit rate of convergence in the $ L^\infty $ norm of the solution, and this improves and extends the previous results.
Keywords: algebraic decay., finite speed propagation, Fourth order, generalized thin film, entropy dissipation method, degenerate parabolic equation.
Mathematics Subject Classification: 35K35, 76A20, 35K55, 35K65, 35B4.
Citation: Lihua Min, Xiaoping Yang. Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 543-566. doi: 10.3934/cpaa.2014.13.543
References:
[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.  Google Scholar

[2]

F. Bernis, Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equations, 1 (1996), 337-368.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Chugunova, M. Pugh and R. Taranets, Research announcement: Finite-time blow up and long-wave unstable thin film equations,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Dal Passo, L. Giacomelli and A. Shishkov, The thin film equation with nonlinear diffusion, Comm. Partial Differential Equations, 26 (2001), 1509-1557.  Google Scholar

[20]

S. D. Èĭdel'man, Parabolic Systems, Translated from the Russian by Scripta Technica, London, North-Holland Publishing Co., Amsterdam, 1969. Google Scholar

[21]

A. Friedman, Partial Differential Equations,, Holt, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

G. Grün, Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal harding, Z. Anal. Anwendungen., 14 (1995), 541-574.  Google Scholar

[25]

G. Grün, "On Free Boundary Problems Arising in Thin Film Flow," Habilitation thesis, University of Bonn, 2001. Google Scholar

[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.  Google Scholar

[27]

L. M. Hocking, Spreading and instability of a viscous fluid sheet, Journal of Fluid Mechanics, 211 (1990), 373-392. doi: 10.1017/S0022112090001616.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

E. O. Tuck and L. W. Schwaxtz, Thin static drops with a free attachment boundary, Journal of Fluid Mechanics, 223 (1991), 313-324.  Google Scholar

[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. Google Scholar

show all references

References:
[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.  Google Scholar

[2]

F. Bernis, Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equations, 1 (1996), 337-368.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Chugunova, M. Pugh and R. Taranets, Research announcement: Finite-time blow up and long-wave unstable thin film equations,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Dal Passo, L. Giacomelli and A. Shishkov, The thin film equation with nonlinear diffusion, Comm. Partial Differential Equations, 26 (2001), 1509-1557.  Google Scholar

[20]

S. D. Èĭdel'man, Parabolic Systems, Translated from the Russian by Scripta Technica, London, North-Holland Publishing Co., Amsterdam, 1969. Google Scholar

[21]

A. Friedman, Partial Differential Equations,, Holt, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

G. Grün, Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal harding, Z. Anal. Anwendungen., 14 (1995), 541-574.  Google Scholar

[25]

G. Grün, "On Free Boundary Problems Arising in Thin Film Flow," Habilitation thesis, University of Bonn, 2001. Google Scholar

[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.  Google Scholar

[27]

L. M. Hocking, Spreading and instability of a viscous fluid sheet, Journal of Fluid Mechanics, 211 (1990), 373-392. doi: 10.1017/S0022112090001616.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

E. O. Tuck and L. W. Schwaxtz, Thin static drops with a free attachment boundary, Journal of Fluid Mechanics, 223 (1991), 313-324.  Google Scholar

[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. Google Scholar

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