Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations

Schimperna, Giulio; Zelik, Sergey

doi:10.1090/S0002-9947-2012-05824-7

Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations
HTML articles powered by AMS MathViewer

by Giulio Schimperna and Sergey Zelik PDF

Trans. Amer. Math. Soc. 365 (2013), 3799-3829 Request permission

Abstract:

A nonlinear parabolic equation of the fourth order is analyzed. The equation is characterized by a mobility coefficient that degenerates at $0$. Existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a priori estimates. If a viscosity term is added and additional conditions on the nonlinear terms are assumed, then it is proved that any weak solution becomes instantaneously strictly positive. This in particular implies uniqueness for strictly positive times and further time-regularization properties. The long-time behavior of the problem is also investigated and the existence of trajectory attractors and, under more restrictive conditions, of strong global attractors is shown.

References

A. V. Babin and M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. MR 1156492
J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997), no. 5, 475–502. MR 1462276, DOI 10.1007/s003329900037
John W. Barrett and James F. Blowey, Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility, Math. Comp. 68 (1999), no. 226, 487–517. MR 1609678, DOI 10.1090/S0025-5718-99-01015-7
John W. Barrett, James F. Blowey, and Harald Garcke, Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Numer. Anal. 37 (1999), no. 1, 286–318. MR 1742748, DOI 10.1137/S0036142997331669
J. Becker and G. Grün, The thin-film equation: recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2005), S291–S307.
Elena Beretta, Michiel Bertsch, and Roberta Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal. 129 (1995), no. 2, 175–200. MR 1328475, DOI 10.1007/BF00379920
Francisco Bernis and Avner Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations 83 (1990), no. 1, 179–206. MR 1031383, DOI 10.1016/0022-0396(90)90074-Y
A. L. Bertozzi, G. Grün, and T. P. Witelski, Dewetting films: bifurcations and concentrations, Nonlinearity 14 (2001), no. 6, 1569–1592. MR 1867093, DOI 10.1088/0951-7715/14/6/309
A. L. Bertozzi and 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), no. 6, 1535–1564. MR 1304438
A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math. 51 (1998), no. 6, 625–661. MR 1611136, DOI 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.3.CO;2-2
F. Brezzi and G. Gilardi, FEM Mathematics, in Finite Element Handbook (H. Kardestuncer Ed.), Part I: Chapt. 1: Functional Analysis, 1.1–1.5; Chapt. 2: Functional Spaces, 2.1–2.11; Chapt. 3: Partial Differential Equations, 3.1–3.6, McGraw-Hill Book Co., New York, 1987.
J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258–267.
Vladimir V. Chepyzhov and Mark I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002. MR 1868930, DOI 10.1051/cocv:2002056
M. I. Vishik, S. V. Zelik, and V. V. Chepyzhov, A strong trajectory attractor for a dissipative reactor-diffusion system, Dokl. Akad. Nauk 435 (2010), no. 2, 155–159 (Russian); English transl., Dokl. Math. 82 (2010), no. 3, 869–873. MR 2790502, DOI 10.1134/S1064562410060086
V. V. Chepyzhov, M. I. Vishik, and S. V. Zelik, Strong trajectory attractors for dissipative Euler equations, J. Math. Pures Appl. (9) 96 (2011), no. 4, 395–407 (English, with English and French summaries). MR 2832641, DOI 10.1016/j.matpur.2011.04.007
Roberta Dal Passo and Harald Garcke, Solutions of a fourth order degenerate parabolic equation with weak initial trace, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 1, 153–181. MR 1679081
Roberta Dal Passo, Lorenzo Giacomelli, and Andrey Shishkov, The thin film equation with nonlinear diffusion, Comm. Partial Differential Equations 26 (2001), no. 9-10, 1509–1557. MR 1865938, DOI 10.1081/PDE-100107451
Arnaud Debussche and Lucia Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal. 24 (1995), no. 10, 1491–1514. MR 1327930, DOI 10.1016/0362-546X(94)00205-V
A. Eden, V. Kalantarov, and S. Zelik, Infinite energy solutions for the Cahn-Hilliard equations in cylindrical domains, submitted.
Charles M. Elliott and Harald Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal. 27 (1996), no. 2, 404–423. MR 1377481, DOI 10.1137/S0036141094267662
C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differential Equations 128 (1996), no. 2, 387–414. MR 1398327, DOI 10.1006/jdeq.1996.0101
Raul Ferreira and Francisco Bernis, Source-type solutions to thin-film equations in higher dimensions, European J. Appl. Math. 8 (1997), no. 5, 507–524. MR 1479525, DOI 10.1017/S0956792597003197
G. Grün, Degenerate parabolic differential equations of fourth order and a plasticity model with non-local hardening, Z. Anal. Anwendungen 14 (1995), no. 3, 541–574. MR 1362530, DOI 10.4171/ZAA/639
Günther Grün, On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions, Math. Comp. 72 (2003), no. 243, 1251–1279. MR 1972735, DOI 10.1090/S0025-5718-03-01492-3
Günther Grün and Martin Rumpf, Simulation of singularities and instabilities arising in thin film flow, European J. Appl. Math. 12 (2001), no. 3, 293–320. The dynamics of thin fluid films. MR 1936040, DOI 10.1017/S0956792501004429
Morton E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D 92 (1996), no. 3-4, 178–192. MR 1387065, DOI 10.1016/0167-2789(95)00173-5
A. D. Ioffe, On lower semicontinuity of integral functionals. I, SIAM J. Control Optim. 15 (1977), no. 4, 521–538. MR 637234, DOI 10.1137/0315035
Ioana Moise, Ricardo Rosa, and Xiaoming Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity 11 (1998), no. 5, 1369–1393. MR 1644413, DOI 10.1088/0951-7715/11/5/012
Alain Miranville and Sergey Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci. 27 (2004), no. 5, 545–582. MR 2041814, DOI 10.1002/mma.464
A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl. 8 (1998), no. 2, 965–985. MR 1657208
Amy Novick-Cohen and Andrey Shishkov, Upper bounds for coarsening for the degenerate Cahn-Hilliard equation, Discrete Contin. Dyn. Syst. 25 (2009), no. 1, 251–272. MR 2525177, DOI 10.3934/dcds.2009.25.251
Amy Novick-Cohen and Andrey Shishkov, The thin film equation with backwards second order diffusion, Interfaces Free Bound. 12 (2010), no. 4, 463–496. MR 2754213, DOI 10.4171/IFB/242
Giulio Schimperna, Global attractors for Cahn-Hilliard equations with nonconstant mobility, Nonlinearity 20 (2007), no. 10, 2365–2387. MR 2356115, DOI 10.1088/0951-7715/20/10/006
Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312, DOI 10.1007/978-1-4612-0645-3
Hao Wu and Songmu Zheng, Global attractor for the 1-D thin film equation, Asymptot. Anal. 51 (2007), no. 2, 101–111. MR 2311155