no. 3
Electrified thin films: Global existence of non-negative solutions
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 413-433

We consider an equation modeling the evolution of a viscous liquid thin film wetting a horizontal solid substrate destabilized by an electric field normal to the substrate. The effects of the electric field are modeled by a lower order non-local term. We introduce the good functional analysis framework to study this equation on a bounded domain and prove the existence of weak solutions defined globally in time for general initial data (with finite energy).

DOI : 10.1016/j.anihpc.2012.01.003
Classification : 35G25, 35K25, 35A01, 35B09
Keywords: Higher order equation, Non-local equation, Thin film equation, Non-negative solutions 
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     author = {Imbert, C. and Mellet, A.},
     title = {Electrified thin films: {Global} existence of non-negative solutions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {413--433},
     year = {2012},
     publisher = {Elsevier},
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}
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Imbert, C.; Mellet, A. Electrified thin films: Global existence of non-negative solutions. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 413-433. doi: 10.1016/j.anihpc.2012.01.003

[1] M.S. Agranovich, B.A. Amosov, On Fourier series in eigenfunctions of elliptic boundary value problems, Georgian Math. J. 10 (2003), 401-410 | EuDML | Zbl | MR

[2] E. Beretta, M. Bertsch, R. Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Ration. Mech. Anal. 129 (1995), 175-200 | Zbl | MR

[3] F. Bernis, A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations 83 (1990), 179-206 | Zbl | MR

[4] A.L. Bertozzi, M. Pugh, The lubrication approximation for thin viscous films: the moving contact line with a “porous media” cut-off of van der Waals interactions, Nonlinearity 7 (1994), 1535-1564 | Zbl | MR

[5] A.L. Bertozzi, 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 | Zbl | MR

[6] A.L. Bertozzi, M.C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math. 51 (1998), 625-661 | MR | Zbl

[7] A.L. Bertozzi, M.C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J. 49 (2000), 1323-1366 | MR | Zbl

[8] X. Cabré, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, preprint, 2009. | MR | Zbl

[9] R. Dal Passo, H. Garcke, 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 | Zbl | MR

[10] M.L. Frankel, On a free boundary problem associated with combustion and solidification, RAIRO Modél. Math. Anal. Numér. 23 (1989), 283-291 | MR | EuDML | Zbl | Numdam

[11] M.L. Frankel, G.I. Sivashinsky, On the equation of a curved flame front, Phys. D 30 (1988), 28-42 | MR | Zbl

[12] G. Grün, Degenerate parabolic differential equations of fourth order and a plasticity model with non-local hardening, Z. Anal. Anwend. 14 (1995), 541-574 | MR | Zbl

[13] G. Grün, On Bernisʼ interpolation inequalities in multiple space dimensions, Z. Anal. Anwend. 20 (2001), 987-998 | MR | Zbl

[14] C. Imbert, A. Mellet, Existence of solutions for a higher order non-local equation appearing in crack dynamics, arXiv:1001.5105 (2010) | MR | Zbl

[15] D.T. Papageorgiou, P.G. Petropoulos, J.-M. Vanden-Broeck, Gravity capillary waves in fluid layers under normal electric fields, Phys. Rev. E (3) 72 (2005) | MR

[16] A.E. Shishkov, R.M. Taranets, On the equation of the flow of thin films with nonlinear convection in multidimensional domains, Ukr. Mat. Visn. 1 (2004), 402-444 | MR | Zbl

[17] J. Simon, Compact sets in the space L p (0,T;B), Ann. Math. Pura Appl. 146 (1987), 65-96 | MR | Zbl

[18] D. Tseluiko, D.T. Papageorgiou, Nonlinear dynamics of electrified thin liquid films, SIAM J. Appl. Math. 67 (2007), 1310-1329 | MR | Zbl

[19] T.P. Witelski, A.J. Bernoff, A.L. Bertozzi, Blowup and dissipation in a critical-case unstable thin film equation, European J. Appl. Math. 15 (2004), 223-256 | MR | Zbl

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