# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [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. [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. [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. [9] E. Carlen and S. Ulusoy, An entropy dissipation-entropy estimate for a thin film type equation, Comm. Math. Sci., 3 (2005), 171-178. [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. [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. [12] S. D. Èĭdel'man, "Parabolic Systems,'' Translated from the Russian by Scripta Technica, London. North-Holland Publishing Co., Amsterdam, 1969. [13] Günther Grün, Droplet spreading under weak slippage: a basic result on finite speed of propagation, SIAM J. Math. Anal., 34 (2003), 992-1006. [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. [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. [16] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1996), 733-737. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [9] E. Carlen and S. Ulusoy, An entropy dissipation-entropy estimate for a thin film type equation, Comm. Math. Sci., 3 (2005), 171-178. [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. [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. [12] S. D. Èĭdel'man, "Parabolic Systems,'' Translated from the Russian by Scripta Technica, London. North-Holland Publishing Co., Amsterdam, 1969. [13] Günther Grün, Droplet spreading under weak slippage: a basic result on finite speed of propagation, SIAM J. Math. Anal., 34 (2003), 992-1006. [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. [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. [16] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1996), 733-737. [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. [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. [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.
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