# 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 & 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. 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 S\'er. I Math., 322 (1996), 1169. Google Scholar [3] Francisco Bernis and Avner Friedman, Higher order nonlinear degenerate parabolic equations,, J. Differential Equations, 83 (1990), 179. 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. 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. 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. 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. 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,'', (1994), 155. 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. 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. 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. doi: doi:10.1006/jcis.1994.1415. Google Scholar [12] S. D. Èĭdel'man, "Parabolic Systems,'', Translated from the Russian by Scripta Technica, (1969). Google Scholar [13] Günther Grün, Droplet spreading under weak slippage: a basic result on finite speed of propagation,, SIAM J. Math. Anal., (2003), 992. Google Scholar [14] Ansgar Jungel and Danial Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs,, Nonlinearity, 19 (2006), 633. 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., (2005), 613. 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. 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. Google Scholar [18] A. Tudorascu, Lubrication approximation for thin viscous films: asymptotic behavior of nonnegative solutions,, Communications in PDE, 32 (2007), 1147. 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. 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. 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 S\'er. I Math., 322 (1996), 1169. Google Scholar [3] Francisco Bernis and Avner Friedman, Higher order nonlinear degenerate parabolic equations,, J. Differential Equations, 83 (1990), 179. 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. 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. 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. 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. 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,'', (1994), 155. 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. 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. 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. doi: doi:10.1006/jcis.1994.1415. Google Scholar [12] S. D. Èĭdel'man, "Parabolic Systems,'', Translated from the Russian by Scripta Technica, (1969). Google Scholar [13] Günther Grün, Droplet spreading under weak slippage: a basic result on finite speed of propagation,, SIAM J. Math. Anal., (2003), 992. Google Scholar [14] Ansgar Jungel and Danial Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs,, Nonlinearity, 19 (2006), 633. 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., (2005), 613. 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. 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. Google Scholar [18] A. Tudorascu, Lubrication approximation for thin viscous films: asymptotic behavior of nonnegative solutions,, Communications in PDE, 32 (2007), 1147. 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. doi: doi:10.1063/1.870138. Google Scholar
 [1] Lili Ju, Xinfeng Liu, Wei Leng. Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667 [2] José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1 [3] Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617 [4] M. Ben Ayed, K. El Mehdi, M. Hammami. Nonexistence of bounded energy solutions for a fourth order equation on thin annuli. Communications on Pure & Applied Analysis, 2004, 3 (4) : 557-580. doi: 10.3934/cpaa.2004.3.557 [5] Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851 [6] Feliz Minhós, João Fialho. On the solvability of some fourth-order equations with functional boundary conditions. Conference Publications, 2009, 2009 (Special) : 564-573. doi: 10.3934/proc.2009.2009.564 [7] Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729 [8] Benoît Pausader. The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1275-1292. doi: 10.3934/dcds.2009.24.1275 [9] Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227 [10] Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225 [11] Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018153 [12] Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781 [13] Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081 [14] José A. Carrillo, Jean Dolbeault, Ivan Gentil, Ansgar Jüngel. Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1027-1050. doi: 10.3934/dcdsb.2006.6.1027 [15] Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks & Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941 [16] A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 [17] Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170 [18] Zhilin Yang, Jingxian Sun. Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1615-1628. doi: 10.3934/cpaa.2012.11.1615 [19] Maria-Magdalena Boureanu. Fourth-order problems with Leray-Lions type operators in variable exponent spaces. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 231-243. doi: 10.3934/dcdss.2019016 [20] Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931

2018 Impact Factor: 0.925