March  2011, 10(2): 803-815. doi: 10.3934/cpaa.2011.10.803

Energy minimizers of a thin film equation with born repulsion force

1. 

Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, United States

Received  January 2010 Revised  May 2010 Published  December 2010

In this paper, we consider a singular elliptic equation modeling steady states of thin film equation with both Van der Waal force and Born repulsion force. We prove the existence of smooth positive energy minimizing solutions. We also investigate the regularity of local minimizers and the limiting behavior of energy minimizer as the Born repulsion force tends to zero.
Citation: Huiqiang Jiang. Energy minimizers of a thin film equation with born repulsion force. Communications on Pure & Applied Analysis, 2011, 10 (2) : 803-815. doi: 10.3934/cpaa.2011.10.803
References:
[1]

R. Almgren, A. Bertozzi and M. Brenner, Stable and unstable singularities in the unforced Hele-Shaw cell,, Phys. Fluids, 8 (1996), 1356. doi: doi:10.1063/1.868915.

[2]

G. Barenblatt, E. Beretta and M. Bertsch, The problem of the spreading of a liquid film along a solid surface: a new mathematical formulation,, Proc. Nat. Acad. Sci. U.S.A., 94 (1997), 10024. doi: doi:10.1073/pnas.94.19.10024.

[3]

E. Beretta, Selfsimilar source solutions of a fourth order degenerate parabolic equation,, Nonlinear Anal., 29 (1997), 741. doi: doi:10.1016/S0362-546X(97)81321-1.

[4]

A. Bernoff and A. 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.

[5]

A. Bertozzi, G. Grün and T. Witelski, Dewetting films: bifurcations and concentrations,, Nonlinearity, 14 (2001), 1569. doi: doi:10.1088/0951-7715/14/6/309.

[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.

[7]

A. Bertozzi and M. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations,, Indiana Univ. Math. J., 49 (2000), 1323.

[8]

A. Bertozzi, The mathematics of moving contact lines in thin liquid films,, Notices Amer. Math. Soc., 45 (1998), 689.

[9]

A. Bertozzi, M. Brenner, T. Dupont and L. Kadanoff, Singularities and similarities in interface flows,, In, (1994), 155.

[10]

M. Bertsch, R. DalPasso, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions,, Adv. Differential Equations, 3 (1998), 417.

[11]

P. Constantin, T. Dupont, R. Goldstein, L. Kadanoff, M. Shelley and S. Zhou, Droplet breakup in a model of the Hele-Shaw cell,, Phys. Rev. E, 47 (1993), 4169. doi: doi:10.1103/PhysRevE.47.4169.

[12]

R. DalPasso, 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. doi: doi:10.1137/S0036141096306170.

[13]

T. Dupont, R. Goldstein, L. Kadanoff and S. Zhou, Finite-time singularity formation in Hele-Shaw systems,, Phys. Rev. E, 47 (1993), 4182. doi: doi:10.1103/PhysRevE.47.4182.

[14]

P. Ehrhard, The spreading of hanging drops,, J. Colloid & Interface, 168 (1994), 242.

[15]

C. Gui, X. Luo and F. Zhou, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations, part ii,, Preprint., ().

[16]

C. Gui and F. Zhou, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations,, Methods Appl. Anal., 15 (2008), 285.

[17]

H. Jiang and F. Lin, Zero set of Sobolev functions with negative power of integrability,, Chinese Ann. Math. Ser. B, 25 (2004), 65. doi: doi:10.1142/S0252959904000068.

[18]

H. Jiang and W. Ni, On steady states of van der Waals force driven thin film equations,, European J. Appl. Math., 18 (2007), 153. doi: doi:10.1017/S0956792507006936.

[19]

R. Laugesen and M. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations,, Arch. Ration. Mech. Anal., 154 (2000), 3. doi: doi:10.1007/PL00004234.

[20]

R. Laugesen and M. Pugh, Properties of steady states for thin film equations,, European J. Appl. Math., 11 (2000), 293. doi: doi:10.1017/S0956792599003794.

[21]

R. Laugesen and M. Pugh, Energy levels of steady states for thin-film-type equations,, J. Differential Equations, 182 (2002), 377. doi: doi:10.1006/jdeq.2001.4108.

[22]

R. Laugesen and M. Pugh, Heteroclinic orbits, mobility parameters and stability for thin film type equations,, Electron. J. Differential Equations, (2002).

[23]

T. Myers, Thin films with high surface tension,, SIAM Rev., 40 (1998), 441. doi: doi:10.1137/S003614459529284X.

[24]

A. Oron, S. Davis and S. Bankoff, Nonlinear theory of film rupture,, Rev. Mod. Phys., 69 (1997), 931. doi: doi:10.1103/RevModPhys.69.931.

[25]

M. Shelley, R. Goldstein and A. Pesci, Topological transitions in Hele-Shaw flow,, In, (1993), 167.

[26]

M. Williams and S. Davis, Nonlinear theory of film rupture,, J. Colloid Interf. Sci., 90 (1982), 220. doi: doi:10.1016/0021-9797(82)90415-5.

[27]

T. Witelski and A. 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.

[28]

T. Witelski and A. Bernoff, Dynamics of three-dimensional thin film rupture,, Phys. D, 147 (2000), 155. doi: doi:10.1016/S0167-2789(00)00165-2.

[29]

W. Zhang and J. Lister, Similarity solutions for van der Waals rupture of a thin film on a solid substrate,, Phys. Fluids, 11 (1999), 2454. doi: doi:10.1063/1.870110.

show all references

References:
[1]

R. Almgren, A. Bertozzi and M. Brenner, Stable and unstable singularities in the unforced Hele-Shaw cell,, Phys. Fluids, 8 (1996), 1356. doi: doi:10.1063/1.868915.

[2]

G. Barenblatt, E. Beretta and M. Bertsch, The problem of the spreading of a liquid film along a solid surface: a new mathematical formulation,, Proc. Nat. Acad. Sci. U.S.A., 94 (1997), 10024. doi: doi:10.1073/pnas.94.19.10024.

[3]

E. Beretta, Selfsimilar source solutions of a fourth order degenerate parabolic equation,, Nonlinear Anal., 29 (1997), 741. doi: doi:10.1016/S0362-546X(97)81321-1.

[4]

A. Bernoff and A. 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.

[5]

A. Bertozzi, G. Grün and T. Witelski, Dewetting films: bifurcations and concentrations,, Nonlinearity, 14 (2001), 1569. doi: doi:10.1088/0951-7715/14/6/309.

[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.

[7]

A. Bertozzi and M. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations,, Indiana Univ. Math. J., 49 (2000), 1323.

[8]

A. Bertozzi, The mathematics of moving contact lines in thin liquid films,, Notices Amer. Math. Soc., 45 (1998), 689.

[9]

A. Bertozzi, M. Brenner, T. Dupont and L. Kadanoff, Singularities and similarities in interface flows,, In, (1994), 155.

[10]

M. Bertsch, R. DalPasso, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions,, Adv. Differential Equations, 3 (1998), 417.

[11]

P. Constantin, T. Dupont, R. Goldstein, L. Kadanoff, M. Shelley and S. Zhou, Droplet breakup in a model of the Hele-Shaw cell,, Phys. Rev. E, 47 (1993), 4169. doi: doi:10.1103/PhysRevE.47.4169.

[12]

R. DalPasso, 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. doi: doi:10.1137/S0036141096306170.

[13]

T. Dupont, R. Goldstein, L. Kadanoff and S. Zhou, Finite-time singularity formation in Hele-Shaw systems,, Phys. Rev. E, 47 (1993), 4182. doi: doi:10.1103/PhysRevE.47.4182.

[14]

P. Ehrhard, The spreading of hanging drops,, J. Colloid & Interface, 168 (1994), 242.

[15]

C. Gui, X. Luo and F. Zhou, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations, part ii,, Preprint., ().

[16]

C. Gui and F. Zhou, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations,, Methods Appl. Anal., 15 (2008), 285.

[17]

H. Jiang and F. Lin, Zero set of Sobolev functions with negative power of integrability,, Chinese Ann. Math. Ser. B, 25 (2004), 65. doi: doi:10.1142/S0252959904000068.

[18]

H. Jiang and W. Ni, On steady states of van der Waals force driven thin film equations,, European J. Appl. Math., 18 (2007), 153. doi: doi:10.1017/S0956792507006936.

[19]

R. Laugesen and M. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations,, Arch. Ration. Mech. Anal., 154 (2000), 3. doi: doi:10.1007/PL00004234.

[20]

R. Laugesen and M. Pugh, Properties of steady states for thin film equations,, European J. Appl. Math., 11 (2000), 293. doi: doi:10.1017/S0956792599003794.

[21]

R. Laugesen and M. Pugh, Energy levels of steady states for thin-film-type equations,, J. Differential Equations, 182 (2002), 377. doi: doi:10.1006/jdeq.2001.4108.

[22]

R. Laugesen and M. Pugh, Heteroclinic orbits, mobility parameters and stability for thin film type equations,, Electron. J. Differential Equations, (2002).

[23]

T. Myers, Thin films with high surface tension,, SIAM Rev., 40 (1998), 441. doi: doi:10.1137/S003614459529284X.

[24]

A. Oron, S. Davis and S. Bankoff, Nonlinear theory of film rupture,, Rev. Mod. Phys., 69 (1997), 931. doi: doi:10.1103/RevModPhys.69.931.

[25]

M. Shelley, R. Goldstein and A. Pesci, Topological transitions in Hele-Shaw flow,, In, (1993), 167.

[26]

M. Williams and S. Davis, Nonlinear theory of film rupture,, J. Colloid Interf. Sci., 90 (1982), 220. doi: doi:10.1016/0021-9797(82)90415-5.

[27]

T. Witelski and A. 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.

[28]

T. Witelski and A. Bernoff, Dynamics of three-dimensional thin film rupture,, Phys. D, 147 (2000), 155. doi: doi:10.1016/S0167-2789(00)00165-2.

[29]

W. Zhang and J. Lister, Similarity solutions for van der Waals rupture of a thin film on a solid substrate,, Phys. Fluids, 11 (1999), 2454. doi: doi:10.1063/1.870110.

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