# American Institute of Mathematical Sciences

November  2014, 34(11): 4537-4553. doi: 10.3934/dcds.2014.34.4537

## Localization, smoothness, and convergence to equilibrium for a thin film equation

 1 Department of Mathematics, Hill Center, Rutgers University, Piscataway, NJ 08854, United States 2 Department of Mathematics, Faculty of Education, Zirve University, Gaziantep, Turkey

Received  April 2013 Revised  February 2014 Published  May 2014

We investigate the long-time behavior of weak solutions to the thin-film type equation $$v_t =(xv - vv_{xxx})_x\ ,$$ which arises in the Hele-Shaw problem. We estimate the rate of convergence of solutions to the Smyth-Hill equilibrium solution, which has the form $\frac{1}{24}(C^2-x^2)^2_+$, in the norm $$|\!|\!| f |\!|\!|_{m,1}^2 = \int_{\mathbb{R}}(1+ |x|^{2m})|f(x)|^2 \, dx + \int_{\mathbb{R}}|f_x(x)|^2 \, dx.$$ We obtain exponential convergence in the $|\!|\!| \cdot |\!|\!|_{m,1}$ norm for all $m$ with $1\leq m< 2$, thus obtaining rates of convergence in norms measuring both smoothness and localization. The localization is the main novelty, and in fact, we show that there is a close connection between the localization bounds and the smoothness bounds: Convergence of second moments implies convergence in the $H^1$ Sobolev norm. We then use methods of optimal mass transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak solutions for which all of the estimates on which our convergence analysis depends may be rigorously derived. Though our main results on convergence can be stated without reference to optimal mass transportation, essential use of this theory is made throughout our analysis.
Citation: Eric A. Carlen, Süleyman Ulusoy. Localization, smoothness, and convergence to equilibrium for a thin film equation. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4537-4553. doi: 10.3934/dcds.2014.34.4537
##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the Wasserstein space of probability measures, Birkhäuser Verlag, Basel, 2005. [2] L. Ansini and L. Giacomelli, Doubly nonlinear thin-film equation in one space dimension, Arch. Ration. Mech. Anal., 173 (2004), 89-131. doi: 10.1007/s00205-004-0313-x. [3] J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, J. Phys.: Condens. Matter, 17 (2005), 291-307. doi: 10.1088/0953-8984/17/9/002. [4] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Eqns., 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. [5] A. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices AMS, 45 (1998), 689-697. [6] A. 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: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. [7] M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Diff. Eqns., 3 (1998), 417-440. [8] M. Boutat, S. Hilout, J. E. Rakotoson and J. M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Anal. TMA, 69 (2008), 1268-1286. doi: 10.1016/j.na.2007.06.028. [9] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402. [10] E. A. Carlen and S. Ulusoy, Asymptotic equipartition and long time behavior of solutions of a thin-film equation, J. Diff. Eqns., 241 (2007), 279-292. doi: 10.1016/j.jde.2007.07.005. [11] J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin-film equation, Comm. Math. Phys., 225 (2002), 551-571. doi: 10.1007/s002200100591. [12] M. Chugunova, M. Pugh and R. M. Taranets, Nonnegative solutions for a long-wave unstable thin film equation with convection, SIAM J. Math. Anal., 42 (2010), 1826-1853. doi: 10.1137/090777062. [13] R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170. [14] L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calc. Var. Part. Diff. Eq., 13 (2001), 377-403. doi: 10.1007/s005260000077. [15] G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Comm. PDEs, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193. [16] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [17] R. Laugesen, New dissipated energies for the thin fluid film equation, Comm. Pure Appl. Analysis, 4 (2005), 613-634. doi: 10.3934/cpaa.2005.4.613. [18] D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. PDEs, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256. [19] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. [20] T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X. [21] F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. PDEs, 23 (1998), 2077-2164. doi: 10.1080/03605309808821411. [22] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. PDEs, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. [23] J. E. Rakotoson, J. M. Rakotoson and C. Verbeke, Higher order equations related to thin films: Blow-up and global existence, the influence of the initial data, J. Diff. Eqns., 244 (2008), 2693-2740. doi: 10.1016/j.jde.2008.03.009. [24] N. F. Smyth and J. M. Hill, High-order nonlinear diffusion, IMA J. Appl. Math., 40 (1988), 73-86. doi: 10.1093/imamat/40.2.73. [25] A. Tudorascu, Lubrication approximation for viscous flows: asyptotic behavior of nonnegative solutions, Comm. PDEs, 32 (2007), 1147-1172. doi: 10.1080/03605300600987272. [26] S. Ulusoy, A new family of higher order nonlinear degenerate parabolic equations, Nonlinearity, 20 (2007), 685-712. doi: 10.1088/0951-7715/20/3/007. [27] S. Ulusoy, On a new family of higher order nonlinear degenerate parabolic equations, Appl. Math. Res. eXpress, 2007 (2007), Article ID abm010, 28 pages. doi: 10.1088/0951-7715/20/3/007. [28] S. Ulusoy, The Mathematical Theory of Thin Film Evolution, Ph.D thesis, Georgia Institute of Technology, 2007. [29] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., 58, AMS, Providence, RI, 2003. doi: 10.1007/b12016. [30] J. L. Vazquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

show all references

##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the Wasserstein space of probability measures, Birkhäuser Verlag, Basel, 2005. [2] L. Ansini and L. Giacomelli, Doubly nonlinear thin-film equation in one space dimension, Arch. Ration. Mech. Anal., 173 (2004), 89-131. doi: 10.1007/s00205-004-0313-x. [3] J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, J. Phys.: Condens. Matter, 17 (2005), 291-307. doi: 10.1088/0953-8984/17/9/002. [4] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Eqns., 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. [5] A. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices AMS, 45 (1998), 689-697. [6] A. 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: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. [7] M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Diff. Eqns., 3 (1998), 417-440. [8] M. Boutat, S. Hilout, J. E. Rakotoson and J. M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Anal. TMA, 69 (2008), 1268-1286. doi: 10.1016/j.na.2007.06.028. [9] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402. [10] E. A. Carlen and S. Ulusoy, Asymptotic equipartition and long time behavior of solutions of a thin-film equation, J. Diff. Eqns., 241 (2007), 279-292. doi: 10.1016/j.jde.2007.07.005. [11] J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin-film equation, Comm. Math. Phys., 225 (2002), 551-571. doi: 10.1007/s002200100591. [12] M. Chugunova, M. Pugh and R. M. Taranets, Nonnegative solutions for a long-wave unstable thin film equation with convection, SIAM J. Math. Anal., 42 (2010), 1826-1853. doi: 10.1137/090777062. [13] R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170. [14] L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calc. Var. Part. Diff. Eq., 13 (2001), 377-403. doi: 10.1007/s005260000077. [15] G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Comm. PDEs, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193. [16] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [17] R. Laugesen, New dissipated energies for the thin fluid film equation, Comm. Pure Appl. Analysis, 4 (2005), 613-634. doi: 10.3934/cpaa.2005.4.613. [18] D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. PDEs, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256. [19] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. [20] T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X. [21] F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. PDEs, 23 (1998), 2077-2164. doi: 10.1080/03605309808821411. [22] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. PDEs, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. [23] J. E. Rakotoson, J. M. Rakotoson and C. Verbeke, Higher order equations related to thin films: Blow-up and global existence, the influence of the initial data, J. Diff. Eqns., 244 (2008), 2693-2740. doi: 10.1016/j.jde.2008.03.009. [24] N. F. Smyth and J. M. Hill, High-order nonlinear diffusion, IMA J. Appl. Math., 40 (1988), 73-86. doi: 10.1093/imamat/40.2.73. [25] A. Tudorascu, Lubrication approximation for viscous flows: asyptotic behavior of nonnegative solutions, Comm. PDEs, 32 (2007), 1147-1172. doi: 10.1080/03605300600987272. [26] S. Ulusoy, A new family of higher order nonlinear degenerate parabolic equations, Nonlinearity, 20 (2007), 685-712. doi: 10.1088/0951-7715/20/3/007. [27] S. Ulusoy, On a new family of higher order nonlinear degenerate parabolic equations, Appl. Math. Res. eXpress, 2007 (2007), Article ID abm010, 28 pages. doi: 10.1088/0951-7715/20/3/007. [28] S. Ulusoy, The Mathematical Theory of Thin Film Evolution, Ph.D thesis, Georgia Institute of Technology, 2007. [29] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., 58, AMS, Providence, RI, 2003. doi: 10.1007/b12016. [30] J. L. Vazquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.
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