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, Süleyman Ulusoy. Localization, smoothness, and convergence to equilibrium for a thin film equation. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4537-4553. doi: 10.3934/dcds.2014.34.4537
References:
[1]

Birkhäuser Verlag, Basel, 2005.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 173 (2004), 89-131. doi: 10.1007/s00205-004-0313-x.  Google Scholar

[3]

J. Phys.: Condens. Matter, 17 (2005), 291-307. doi: 10.1088/0953-8984/17/9/002.  Google Scholar

[4]

J. Diff. Eqns., 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y.  Google Scholar

[5]

Notices AMS, 45 (1998), 689-697.  Google Scholar

[6]

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.  Google Scholar

[7]

Adv. Diff. Eqns., 3 (1998), 417-440.  Google Scholar

[8]

Nonlinear Anal. TMA, 69 (2008), 1268-1286. doi: 10.1016/j.na.2007.06.028.  Google Scholar

[9]

Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402.  Google Scholar

[10]

J. Diff. Eqns., 241 (2007), 279-292. doi: 10.1016/j.jde.2007.07.005.  Google Scholar

[11]

Comm. Math. Phys., 225 (2002), 551-571. doi: 10.1007/s002200100591.  Google Scholar

[12]

SIAM J. Math. Anal., 42 (2010), 1826-1853. doi: 10.1137/090777062.  Google Scholar

[13]

SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170.  Google Scholar

[14]

Calc. Var. Part. Diff. Eq., 13 (2001), 377-403. doi: 10.1007/s005260000077.  Google Scholar

[15]

Comm. PDEs, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193.  Google Scholar

[16]

SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.  Google Scholar

[17]

Comm. Pure Appl. Analysis, 4 (2005), 613-634. doi: 10.3934/cpaa.2005.4.613.  Google Scholar

[18]

Comm. PDEs, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256.  Google Scholar

[19]

Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.  Google Scholar

[20]

SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X.  Google Scholar

[21]

Comm. PDEs, 23 (1998), 2077-2164. doi: 10.1080/03605309808821411.  Google Scholar

[22]

Comm. PDEs, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar

[23]

J. Diff. Eqns., 244 (2008), 2693-2740. doi: 10.1016/j.jde.2008.03.009.  Google Scholar

[24]

IMA J. Appl. Math., 40 (1988), 73-86. doi: 10.1093/imamat/40.2.73.  Google Scholar

[25]

Comm. PDEs, 32 (2007), 1147-1172. doi: 10.1080/03605300600987272.  Google Scholar

[26]

Nonlinearity, 20 (2007), 685-712. doi: 10.1088/0951-7715/20/3/007.  Google Scholar

[27]

Appl. Math. Res. eXpress, 2007 (2007), Article ID abm010, 28 pages. doi: 10.1088/0951-7715/20/3/007.  Google Scholar

[28]

Ph.D thesis, Georgia Institute of Technology, 2007.  Google Scholar

[29]

Grad. Stud. Math., 58, AMS, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

[30]

Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar

show all references

References:
[1]

Birkhäuser Verlag, Basel, 2005.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 173 (2004), 89-131. doi: 10.1007/s00205-004-0313-x.  Google Scholar

[3]

J. Phys.: Condens. Matter, 17 (2005), 291-307. doi: 10.1088/0953-8984/17/9/002.  Google Scholar

[4]

J. Diff. Eqns., 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y.  Google Scholar

[5]

Notices AMS, 45 (1998), 689-697.  Google Scholar

[6]

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.  Google Scholar

[7]

Adv. Diff. Eqns., 3 (1998), 417-440.  Google Scholar

[8]

Nonlinear Anal. TMA, 69 (2008), 1268-1286. doi: 10.1016/j.na.2007.06.028.  Google Scholar

[9]

Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402.  Google Scholar

[10]

J. Diff. Eqns., 241 (2007), 279-292. doi: 10.1016/j.jde.2007.07.005.  Google Scholar

[11]

Comm. Math. Phys., 225 (2002), 551-571. doi: 10.1007/s002200100591.  Google Scholar

[12]

SIAM J. Math. Anal., 42 (2010), 1826-1853. doi: 10.1137/090777062.  Google Scholar

[13]

SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170.  Google Scholar

[14]

Calc. Var. Part. Diff. Eq., 13 (2001), 377-403. doi: 10.1007/s005260000077.  Google Scholar

[15]

Comm. PDEs, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193.  Google Scholar

[16]

SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.  Google Scholar

[17]

Comm. Pure Appl. Analysis, 4 (2005), 613-634. doi: 10.3934/cpaa.2005.4.613.  Google Scholar

[18]

Comm. PDEs, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256.  Google Scholar

[19]

Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634.  Google Scholar

[20]

SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X.  Google Scholar

[21]

Comm. PDEs, 23 (1998), 2077-2164. doi: 10.1080/03605309808821411.  Google Scholar

[22]

Comm. PDEs, 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar

[23]

J. Diff. Eqns., 244 (2008), 2693-2740. doi: 10.1016/j.jde.2008.03.009.  Google Scholar

[24]

IMA J. Appl. Math., 40 (1988), 73-86. doi: 10.1093/imamat/40.2.73.  Google Scholar

[25]

Comm. PDEs, 32 (2007), 1147-1172. doi: 10.1080/03605300600987272.  Google Scholar

[26]

Nonlinearity, 20 (2007), 685-712. doi: 10.1088/0951-7715/20/3/007.  Google Scholar

[27]

Appl. Math. Res. eXpress, 2007 (2007), Article ID abm010, 28 pages. doi: 10.1088/0951-7715/20/3/007.  Google Scholar

[28]

Ph.D thesis, Georgia Institute of Technology, 2007.  Google Scholar

[29]

Grad. Stud. Math., 58, AMS, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

[30]

Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar

[1]

Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3389-3414. doi: 10.3934/dcds.2021001

[2]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012

[3]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[4]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032

[5]

Lifen Jia, Wei Dai. Uncertain spring vibration equation. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021073

[6]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153

[7]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[8]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[9]

Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029

[10]

Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021015

[11]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002

[12]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[13]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[14]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[15]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186

[16]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384

[17]

Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021065

[18]

José A. Carrillo, Bertram Düring, Lisa Maria Kreusser, Carola-Bibiane Schönlieb. Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3985-4012. doi: 10.3934/dcds.2021025

[19]

Samira Shahsavari, Saeed Ketabchi. The proximal methods for solving absolute value equation. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 449-460. doi: 10.3934/naco.2020037

[20]

Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]