THE CAHN-HILLIARD EQUATION WITH SINGULAR POTENTIALS AND DYNAMIC BOUNDARY CONDITIONS

Our aim in this paper is to study the Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. In particular, we prove, owing to proper approximations of the singular potential and a suitable notion of variational solutions, the existence and uniqueness of solutions. We also discuss the separation of the solutions from the singularities of the potential. Finally, we prove the existence of global and exponential attractors.

[1]  Songmu Zheng,et al.  The Cahn-Hilliard equation with dynamic boundary conditions , 2003, Advances in Differential Equations.

[2]  Josef Málek,et al.  Large Time Behavior via the Method of ℓ-Trajectories , 2002 .

[3]  Ciprian G. Gal,et al.  A Cahn–Hilliard model in bounded domains with permeable walls , 2006 .

[4]  John W. Cahn,et al.  On Spinodal Decomposition , 1961 .

[5]  F. Otto,et al.  Upper Bounds on Coarsening Rates , 2002 .

[6]  Thomas Wanner,et al.  Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics , 2000 .

[7]  A. Miranville,et al.  On the Caginalp system with dynamic boundary conditions and singular potentials , 2009 .

[8]  N. Kenmochi,et al.  Subdifferential Operator Approach to the Cahn-Hilliard Equation with Constraint , 1995 .

[9]  Philipp Maass,et al.  Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall , 1998 .

[10]  Sergey Zelik,et al.  Finite- and infinite-dimensional attractors for porous media equations , 2008 .

[11]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[12]  Sergey Zelik,et al.  Finite‐dimensional attractors and exponential attractors for degenerate doubly nonlinear equations , 2009 .

[13]  Charles M. Elliott,et al.  The Cahn-Hilliard Model for the Kinetics of Phase Separation , 1989 .

[14]  Sergey Zelik,et al.  Exponential attractors for a nonlinear reaction-diffusion system in ? , 2000 .

[15]  Thomas Wanner,et al.  Spinodal Decomposition for the Cahn–Hilliard Equation in Higher Dimensions.¶Part I: Probability and Wavelength Estimate , 1998 .

[16]  H. Gajewski,et al.  The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity , 2002, Advances in Differential Equations.

[17]  Jan Prüss,et al.  Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions , 2006 .

[18]  Philipp Maass,et al.  Novel Surface Modes in Spinodal Decomposition , 1997 .

[19]  Philipp Maass,et al.  Diverging time and length scales of spinodal decomposition modes in thin films , 1998 .

[20]  Sergey Zelik,et al.  Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions , 2005 .

[21]  James S. Langer,et al.  Theory of spinodal decomposition in alloys , 1971 .

[22]  Bernd Rinn,et al.  Phase separation in confined geometries: Solving the Cahn–Hilliard equation with generic boundary conditions , 2001 .

[23]  G. Schimperna.  Weak solution to a phase‐field transmission problem in a concentrated capacity , 1999 .

[24]  Sergey Zelik,et al.  Robust exponential attractors for Cahn‐Hilliard type equations with singular potentials , 2004 .

[25]  P. Sheng,et al.  A variational approach to moving contact line hydrodynamics , 2006, Journal of Fluid Mechanics.

[26]  R. Chill,et al.  Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions , 2003 .

[27]  Desheng Li,et al.  Global Attractor for the Cahn–Hilliard System with Fast Growing Nonlinearity , 1998 .

[28]  S. Gatti,et al.  Corrigendum to Existence of global solutions to the Caginalp phase-field system with dynamic bounda , 2008 .

[29]  A. Eden,et al.  Exponential Attractors for Dissipative Evolution Equations , 1995 .

[30]  M. Vishik,et al.  Attractors of Evolution Equations , 1992 .

[31]  Stig Larsson,et al.  THE CAHN-HILLIARD EQUATION , 2007 .

[32]  Roger Temam,et al.  Some Global Dynamical Properties of a Class of Pattern Formation Equations , 1989 .

[33]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[34]  Helmut Abels,et al.  Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy , 2007 .

[35]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[36]  Arnaud Debussche,et al.  On the Cahn-Hilliard equation with a logarithmic free energy , 1995 .

[37]  Giuseppe Savaré,et al.  Variational convergence of nonlinear diffusion equations: applications to concentrated capacity problems with change of phase , 1997 .

[38]  C. M. Elliott,et al.  On the Cahn-Hilliard equation with degenerate mobility , 1996 .

[39]  Sergey Zelik,et al.  Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems* , 2005, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[40]  Alain Miranville,et al.  On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions , 2009 .

[41]  K. Binder,et al.  Dynamics of surface enrichment: A theory based on the Kawasaki spin-exchange model in the presence of a wall , 1991 .

[42]  C. M. Elliott,et al.  The viscous Cahn-Hilliard equation , 2002 .

[43]  Piotr Rybka,et al.  Convergence of solutions to cahn-hilliard equation , 1999 .

[44]  S. Gatti,et al.  Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials , 2008 .

[45]  Energy methods for the Cahn−Hilliard equation , 1988 .

[46]  Sergey Zelik,et al.  Chapter 3 Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains , 2008 .