Electrified thin films: global existence of non-negative solutions. (English) Zbl 1308.35123
Summary: 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).
MSC:
| 35K35 | Initial-boundary value problems for higher-order parabolic equations |
| 74K35 | Thin films |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B09 | Positive solutions to PDEs |
References:
| [1] | Agranovich, M. S.; Amosov, B. A., On Fourier series in eigenfunctions of elliptic boundary value problems, Georgian Math. J., 10, 401-410 (2003), dedicated to the 100th birthday anniversary of Professor Victor Kupradze · Zbl 1050.35054 |
| [2] | Beretta, E.; Bertsch, M.; Dal Passo, R., Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Ration. Mech. Anal., 129, 175-200 (1995) · Zbl 0827.35065 |
| [3] | Bernis, F.; Friedman, A., Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83, 179-206 (1990) · Zbl 0702.35143 |
| [4] | Bertozzi, A. L.; Pugh, M., The lubrication approximation for thin viscous films: the moving contact line with a “porous media” cut-off of van der Waals interactions, Nonlinearity, 7, 1535-1564 (1994) · Zbl 0811.35045 |
| [5] | Bertozzi, A. L.; Pugh, M., The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions, Comm. Pure Appl. Math., 49, 85-123 (1996) · Zbl 0863.76017 |
| [6] | Bertozzi, A. L.; Pugh, M. C., Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51, 625-661 (1998) · Zbl 0916.35008 |
| [7] | Bertozzi, A. L.; Pugh, M. C., Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49, 1323-1366 (2000) · Zbl 0978.35007 |
| [8] | X. Cabré, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, preprint, 2009.; X. Cabré, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, preprint, 2009. |
| [9] | Dal Passo, R.; Garcke, H.; Grün, G., On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29, 321-342 (1998), (electronic) · Zbl 0929.35061 |
| [10] | Frankel, M. L., On a free boundary problem associated with combustion and solidification, RAIRO Modél. Math. Anal. Numér., 23, 283-291 (1989) · Zbl 0669.65089 |
| [11] | Frankel, M. L.; Sivashinsky, G. I., On the equation of a curved flame front, Phys. D, 30, 28-42 (1988) · Zbl 0695.35210 |
| [12] | Grün, G., Degenerate parabolic differential equations of fourth order and a plasticity model with non-local hardening, Z. Anal. Anwend., 14, 541-574 (1995) · Zbl 0835.35061 |
| [13] | Grün, G., On Bernisʼ interpolation inequalities in multiple space dimensions, Z. Anal. Anwend., 20, 987-998 (2001) · Zbl 0996.35026 |
| [14] | Imbert, C.; Mellet, A., Existence of solutions for a higher order non-local equation appearing in crack dynamics (2010) · Zbl 1230.35053 |
| [15] | Papageorgiou, D. T.; Petropoulos, P. G.; Vanden-Broeck, J.-M., Gravity capillary waves in fluid layers under normal electric fields, Phys. Rev. E (3), 72 (2005), 051601, 9 pp |
| [16] | Shishkov, A. E.; Taranets, R. M., On the equation of the flow of thin films with nonlinear convection in multidimensional domains, Ukr. Mat. Visn., 1, 402-444 (2004), 447 · Zbl 1211.35137 |
| [17] | Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. Math. Pura Appl., 146, 65-96 (1987) · Zbl 0629.46031 |
| [18] | Tseluiko, D.; Papageorgiou, D. T., Nonlinear dynamics of electrified thin liquid films, SIAM J. Appl. Math., 67, 1310-1329 (2007), (electronic) · Zbl 1344.76095 |
| [19] | Witelski, T. P.; Bernoff, A. J.; Bertozzi, A. L., Blowup and dissipation in a critical-case unstable thin film equation, European J. Appl. Math., 15, 223-256 (2004) · Zbl 1062.76005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
