March  2011, 10(2): 613-624. doi: 10.3934/cpaa.2011.10.613

New dissipated energy for the unstable thin film equation

1. 

University of Toronto, Department of Mathematics, 40 St. George Str., Toronto, Ontario M5S 2E4, Canada

2. 

Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, 74 R. Luxemburg Str., Donetsk, 83114, Ukraine

Received  February 2010 Revised  August 2010 Published  December 2010

The fluid thin film equation $h_t = - (h^n h_{x x x})_x - a_1 (h^m h_x)_x$ is known to conserve mass $\int h dx$, and in the case of $a_1 \leq 0$, to dissipate entropy $\int h^{3/2 - n} dx$ (see [8]) and the $L^2$-norm of the gradient $\int h_x^2 dx$ (see [3]). For the special case of $a_1 = 0$ a new dissipated quantity $\int h^{\alpha} h_x^2 dx $ was recently discovered for positive classical solutions by Laugesen (see [15]). We extend it in two ways. First, we prove that Laugesen's functional dissipates strong nonnegative generalized solutions. Second, we prove the full $\alpha$-energy $\int (\frac{1}{2} h^\alpha h_x^2 - $ $ \frac {a_1 h^{\alpha + m - n + 2}}{(\alpha + m - n + 1)(\alpha + m - n + 2)} ) dx $ dissipation for strong nonnegative generalized solutions in the case of the unstable porous media perturbation $a_1> 0$ and the critical exponent $m = n+2$.
Citation: Marina Chugunova, Roman M. Taranets. New dissipated energy for the unstable thin film equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 613-624. doi: 10.3934/cpaa.2011.10.613
References:
[1]

Elena Beretta, Michiel Bertsch and Roberta Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal., 129 (1995), 175-200. doi: doi:10.1007/BF00379920.  Google Scholar

[2]

Francisco Bernis, Finite speed of propagation for thin viscous flows when $2\leq n<3$, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 1169-1174.  Google Scholar

[3]

Francisco Bernis and Avner Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206. doi: doi:10.1016/0022-0396(90)90074-Y.  Google Scholar

[4]

Andrew J. Bernoff and Andrea L. Bertozzi, Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition, Phys. D, 85 (1995), 375-404. doi: doi:10.1016/0167-2789(95)00054-8.  Google Scholar

[5]

A. L. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123. doi: doi:10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2.  Google Scholar

[6]

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: doi:10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9.  Google Scholar

[7]

A. L. Bertozzi and M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366. doi: doi:10.1512/iumj.2000.49.1887.  Google Scholar

[8]

Andrea L. Bertozzi, Michael P. Brenner, Todd F. Dupont and Leo P. Kadanoff, Singularities and similarities in interface flows, "Trends and Perspectives in Applied Mathematics,'' volume 100 of Appl. Math. Sci., pages 155-208. Springer, New York, 1994.  Google Scholar

[9]

E. Carlen and S. Ulusoy, An entropy dissipation-entropy estimate for a thin film type equation, Comm. Math. Sci., 3 (2005), 171-178.  Google Scholar

[10]

P. Constantin, T. F. Dupont, R. E. Goldstein, Leo P. Kadanoff, M. J. Shelley and S. M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Physical Review E, 47 (1993), 4169-4181. doi: doi:10.1103/PhysRevE.47.4169.  Google Scholar

[11]

P. Ehrhard, The spreading of hanging drops, Journal of Colloid and Interface Science, 168 (1994), 242-246. doi: doi:10.1006/jcis.1994.1415.  Google Scholar

[12]

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

[13]

Günther Grün, Droplet spreading under weak slippage: a basic result on finite speed of propagation, SIAM J. Math. Anal., 34 (2003), 992-1006.  Google Scholar

[14]

Ansgar Jungel and Danial Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659. doi: doi:10.1088/0951-7715/19/3/006.  Google Scholar

[15]

R. S. Laugesen, New dissipated energies for the thin fluid film equation, Commun. Pure Appl. Anal., 4 (2005), 613-634. doi: doi:10.3934/cpaa.2005.4.613.  Google Scholar

[16]

L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1996), 733-737.  Google Scholar

[17]

A. E. Shishkov and R. M. Taranets., On the equation of the flow of thin films with nonlinear convection in multidimensional domains, Ukr. Mat. Visn., 1 (2004), 402-444.  Google Scholar

[18]

A. Tudorascu, Lubrication approximation for thin viscous films: asymptotic behavior of nonnegative solutions, Communications in PDE, 32 (2007), 1147-1172. doi: doi:10.1080/03605300600987272.  Google Scholar

[19]

Thomas P. Witelski and Andrew J. Bernoff, Stability of self-similar solutions for van der Waals driven thin film rupture, Phys. Fluids, 11 (1999), 2443-2445. doi: doi:10.1063/1.870138.  Google Scholar

show all references

References:
[1]

Elena Beretta, Michiel Bertsch and Roberta Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal., 129 (1995), 175-200. doi: doi:10.1007/BF00379920.  Google Scholar

[2]

Francisco Bernis, Finite speed of propagation for thin viscous flows when $2\leq n<3$, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 1169-1174.  Google Scholar

[3]

Francisco Bernis and Avner Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206. doi: doi:10.1016/0022-0396(90)90074-Y.  Google Scholar

[4]

Andrew J. Bernoff and Andrea L. Bertozzi, Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition, Phys. D, 85 (1995), 375-404. doi: doi:10.1016/0167-2789(95)00054-8.  Google Scholar

[5]

A. L. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123. doi: doi:10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2.  Google Scholar

[6]

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: doi:10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9.  Google Scholar

[7]

A. L. Bertozzi and M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366. doi: doi:10.1512/iumj.2000.49.1887.  Google Scholar

[8]

Andrea L. Bertozzi, Michael P. Brenner, Todd F. Dupont and Leo P. Kadanoff, Singularities and similarities in interface flows, "Trends and Perspectives in Applied Mathematics,'' volume 100 of Appl. Math. Sci., pages 155-208. Springer, New York, 1994.  Google Scholar

[9]

E. Carlen and S. Ulusoy, An entropy dissipation-entropy estimate for a thin film type equation, Comm. Math. Sci., 3 (2005), 171-178.  Google Scholar

[10]

P. Constantin, T. F. Dupont, R. E. Goldstein, Leo P. Kadanoff, M. J. Shelley and S. M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Physical Review E, 47 (1993), 4169-4181. doi: doi:10.1103/PhysRevE.47.4169.  Google Scholar

[11]

P. Ehrhard, The spreading of hanging drops, Journal of Colloid and Interface Science, 168 (1994), 242-246. doi: doi:10.1006/jcis.1994.1415.  Google Scholar

[12]

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

[13]

Günther Grün, Droplet spreading under weak slippage: a basic result on finite speed of propagation, SIAM J. Math. Anal., 34 (2003), 992-1006.  Google Scholar

[14]

Ansgar Jungel and Danial Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659. doi: doi:10.1088/0951-7715/19/3/006.  Google Scholar

[15]

R. S. Laugesen, New dissipated energies for the thin fluid film equation, Commun. Pure Appl. Anal., 4 (2005), 613-634. doi: doi:10.3934/cpaa.2005.4.613.  Google Scholar

[16]

L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1996), 733-737.  Google Scholar

[17]

A. E. Shishkov and R. M. Taranets., On the equation of the flow of thin films with nonlinear convection in multidimensional domains, Ukr. Mat. Visn., 1 (2004), 402-444.  Google Scholar

[18]

A. Tudorascu, Lubrication approximation for thin viscous films: asymptotic behavior of nonnegative solutions, Communications in PDE, 32 (2007), 1147-1172. doi: doi:10.1080/03605300600987272.  Google Scholar

[19]

Thomas P. Witelski and Andrew J. Bernoff, Stability of self-similar solutions for van der Waals driven thin film rupture, Phys. Fluids, 11 (1999), 2443-2445. doi: doi:10.1063/1.870138.  Google Scholar

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