Abstract
For multidimensional equations of flow of thin capillary films with nonlinear diffusion and convection, we prove the existence of a strong nonnegative generalized solution of the Cauchy problem with initial function in the form of a nonnegative Radon measure with compact support. We determine the exact upper estimate (global in time) for the rate of propagation of the support of this solution. The cases where the degeneracy of the equation corresponds to the conditions of “strong” and “weak” slip are analyzed separately. In particular, in the case of “ weak” slip, we establish the exact estimate of decrease in the L 2-norm of the gradient of solution. It is well known that this estimate is not true for the initial functions with noncompact supports.
Similar content being viewed by others
References
F. Bernis, “Viscous flows, fourth order nonlinear degenerate parabolic equations, and singular elliptic problems,” in: J. I. Diaz, M. A. Herrero, A. Linan, and J. L. Vazquez (editors) Free Boundary Problems: Theory and Applications, Pitman Res. Notes Math., Vol. 323, Longman, Harlow (1995) pp. 40–56.
A. L. Bertozzi, A. Munch, and M. Shearer, “Undercompressive shocks in thin film flows,” Physica D, 134, 431–464 (1999).
A. L. Bertozzi and M. Pugh, “The lubrication approximation for thin viscous films: the moving contact line with a porous media cutoff of the Van-der-Waals interactions,” Nonlinearity, 7, 1535–1564 (1994).
A. L. Bertozzi and M. Pugh, “Long-wave instabilities and saturation in thin film equations,” Communs Pure Appl. Math., 51, No. 6, 625–661 (1998).
C. M. Elliot and H. Garcke, “On the Cahn-Hilliard equation with degenerate mobility,” SIAM J. Math. Anal., 27, No. 2, 404–423 (1996).
G. Grün, “Degenerate parabolic differential equations of fourth order and plasticity model with nonlocal hardening,” Z. Anal. Anwendungen., 14, 541–574 (1995).
A. L. Bertozzi, A. Munch, M. Shearer, and K. Zumbrun, “Stability of compressive and undercompressive thin film traveling waves. The dynamics of thin fluid films,” Eur. J. Appl. Math., 12, No. 3, 253–291 (2001).
A. L. Bertozzi and M. Shearer, “Existence of undercompressive traveling waves in thin film equations, ” SIAM J. Math. Anal., 32, 194–213 (2000).
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linè aires, Gauthier-Villars, Dunod (1969).
F. Bernis and A. Friedman, “Higher order nonlinear degenerate parabolic equations,” J. Different. Equat., 83, 179–206 (1990).
F. Bernis, “Finite speed of propagation and continuity of the interface for thin viscous flows, ” Adv. Different. Equat., 1, No. 3, 337–368 (1996).
R. Kersner and A. Shishkov, Existence of Free Boundaries in Thin-Film Theory, Preprint No. 6, Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk (1996).
E. Beretta, M. Bertsch, and R. dal Passo, “Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation,” Arch. Rat. Mech. Anal., 129, No. 2, 175–200 (1995).
F. Bernis, “Finite speed of propagation for thin viscous flows when 2 ≤ n < 3,” C. R. Acad. Sci. Ser. Math., 322, 1169–1174 (1996).
J. Hulshof and A. Shishkov, “The thin film equation with 2 ≤ n < 3: Finite speed of propagation in terms of the L 1-norm,” Adv. Different. Equat., 3, 625–642 (1998).
F. Bernis, L. A. Peletier, and S. M. Williams, “Source type solutions of a fourth-order nonlinear degenerate parabolic equation,” Nonlin. Anal., 18, 217–234 (1992).
F. Bernis and R. Ferreira, “Source-type solutions to thin-film equations in higher dimensions, ” Eur. J. Appl. Math., 8, 507–524 (1997).
E. Beretta, “Self-similar source solutions of a fourth-order degenerate parabolic equation,” Nonlin. Anal., 29, No. 7, 741–760 (1997).
A. L. Bertozzi and M. Pugh, “The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions,” Comm. Pure Appl. Math., 49, No. 2, 85–123 (1994).
L. Giacomelli, “A fourth-order degenerate parabolic equation describing thin viscous flows over an inclined plane,” Appl. Math. Lett., 12, No. 8, 107–111 (1999).
L. Giacomelli and A. Shishkov, “Propagation of support in one-dimensional convected thin-film flow, ” Indiana Univ. Math. J., 54, No. 4, 1181–1215 (2005).
R. dal Passo, H. Garcke, and G. Grun, “On a fourth-order degenerate parabolic equation: Global entropy estimates, existence and qualitative behavior of solutions,” SIAM J. Math. Anal., 29, No. 2, 321–342 (1998).
G. Grün, On Free-Boundary Problems Arising in Thin-Film Flow, Habilitat. Thesis, Univ. Bonn (2001) (accepted).
R. dal Passo, L. Giacomelli, and A. Shishkov, “The thin film equation with nonlinear diffusion, ” Comm. Part. Different. Equat., 26, 1509–1557_(2001).
R. M. Taranets and A. E. Shishkov, “On the equation of flow of thin films with nonlinear convection in multidimensional domains,” Ukr. Mat. Visn., 1, No. 3, 402–444 (2004).
M. Bertsch, R. dal Passo, H. Garcke, and G. Grun, “The thin viscous flow equation in higher space dimension,” Adv. Different. Equat., 3, 417–440 (1998).
R. dal Passo and H. Garcke, “Solutions of a fourth-order degenerate parabolic equation with weak initial trace,” Ann. S. N. S., Classe Sci., 28, 153–181 (1999).
R. M. Taranets, Solvability and Qualitative Properties of Solutions of the Equations of Thin Capillary Films with Nonlinear Diffusion, Convection, and Absorption [in Russian], Candidate-Degree Thesis (Physics and Mathematics), Donetsk (2005).
L. Nirenberg, “An extended interpolation inequality,” Ann. Scuola Norm. Super. Pisa, 20, 733–737 (1966).
G. Stampacchia, Equations Elliptiques du Second Order a Coefficients Discontinues, Montreal, Univ. Montreal (1966).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 250–271, February, 2006.
Rights and permissions
About this article
Cite this article
Taranets, R.M., Shishkov, A.E. Singular Cauchy problem for the equation of flow of thin viscous films with nonlinear convection. Ukr Math J 58, 280–303 (2006). https://doi.org/10.1007/s11253-006-0066-9
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-006-0066-9