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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[8]

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

[9]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[8]

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

[9]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

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

[2]

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

[3]

Richard S. Laugesen. New dissipated energies for the thin fluid film equation. Communications on Pure & Applied Analysis, 2005, 4 (3) : 613-634. doi: 10.3934/cpaa.2005.4.613

[4]

Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465

[5]

Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070

[6]

Daniel Ginsberg, Gideon Simpson. Analytical and numerical results on the positivity of steady state solutions of a thin film equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1305-1321. doi: 10.3934/dcdsb.2013.18.1305

[7]

Lihua Min, Xiaoping Yang. Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 543-566. doi: 10.3934/cpaa.2014.13.543

[8]

Sergey Degtyarev. Classical solvability of the multidimensional free boundary problem for the thin film equation with quadratic mobility in the case of partial wetting. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3625-3699. doi: 10.3934/dcds.2017156

[9]

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

[10]

Cheng Wang, Xiaoming Wang, Steven M. Wise. Unconditionally stable schemes for equations of thin film epitaxy. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 405-423. doi: 10.3934/dcds.2010.28.405

[11]

P. Álvarez-Caudevilla, J. D. Evans, V. A. Galaktionov. The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 807-827. doi: 10.3934/dcds.2015.35.807

[12]

Hyung Ju Hwang, Thomas P. Witelski. Short-time pattern formation in thin film equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 867-885. doi: 10.3934/dcds.2009.23.867

[13]

Lei Yang, Xiao-Ping Wang. Dynamics of domain wall in thin film driven by spin current. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1251-1263. doi: 10.3934/dcdsb.2010.14.1251

[14]

Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083

[15]

Zongming Guo, Long Wei. A fourth order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2493-2508. doi: 10.3934/cpaa.2014.13.2493

[16]

A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253

[17]

Andrey Shishkov. Waiting time of propagation and the backward motion of interfaces in thin-film flow theory. Conference Publications, 2007, 2007 (Special) : 938-945. doi: 10.3934/proc.2007.2007.938

[18]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071

[19]

Minhajul, T. Raja Sekhar, G. P. Raja Sekhar. Stability of solutions to the Riemann problem for a thin film model of a perfectly soluble anti-surfactant solution. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3367-3386. doi: 10.3934/cpaa.2019152

[20]

Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]