Advanced Search
Article Contents
Article Contents

On a class of sixth order viscous Cahn-Hilliard type equations

Abstract Related Papers Cited by
  • An initial-boundary-value problem for a class of sixth order viscous Cahn-Hilliard type equations with a nonlinear diffusion is considered. The study is motivated by phase-field modelling of various spatial structures, for example arising in oil-water-surfactant mixtures and in modelling of crystal growth on atomic length, known as phase field crystal model. For such problem we prove the existence and uniqueness of a global in time regular solution. First the finite-time existence is proved by means of the Leray-Schauder fixed point theorem. Then, due to suitable estimates, the finite-time solution is extended step by step on the infinite time interval.
    Mathematics Subject Classification: Primary: 35K60; Secondary: 35Q72, 35L205.


    \begin{equation} \\ \end{equation}
  • [1]

    J. Berry, K. R. Elder and M. Grant, Simulation of an atomistic dynamic field theory for monatomic liquids: Freezing and glass formation, Phys. Rev. E, 77 (2008), 061506.doi: 10.1103/PhysRevE.77.061506.


    J. Berry, M. Grant and K. R. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E, 73 (2006), 031609.doi: 10.1103/PhysRevE.73.031609.


    O. V. Besov, V. P. Il'in and S. M. Nikolskij, "Integral Representation of Functions and Theorems of Imbeddings," Nauka, Moscow, 1975 (in Russian).


    D. G. B. Edelen, On the existence of symmetry relations and dissipation potentials, Arch. Ration. Mech. Anal., 51 (1973), 218-227.doi: 10.1007/BF00276075.


    M. Efendiev and A. Miranville, New models of Cahn-Hilliard-Gurtin equations, Continuum Mech. Thermodyn, 16 (2004), 441-451.doi: 10.1007/s00161-003-0169-6.


    K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E., 70 (2004), 051605.doi: 10.1103/PhysRevE.70.051605.


    K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88, (2002), 245701.doi: 10.1103/PhysRevLett.88.245701.


    P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 051110(11).doi: 10.1103/PhysRevE.79.051110.


    G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E, 47 (1993), 4289-4300.doi: 10.1103/PhysRevE.47.4289.


    G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E, 47 (1993), 4301-4312.doi: 10.1103/PhysRevE.47.4301.


    G. Gompper and S. Zschocke, Ginzburg-Landau theory of oil-water-surfactant mixtures, Phys. Rev. A, 46 (1992), 4836-4851.doi: 10.1103/PhysRevA.46.4836.


    M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.doi: 10.1016/0167-2789(95)00173-5.


    M. D. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation, to appear, 2011.


    M. D. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation, to appear, 2011.


    I. S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Rational Mech. Anal., 46 (1972), 131-148.doi: 10.1007/BF00250688.


    I. Müller, "Thermodynamics," Pitman, London, 1985.doi: 10.1097/00006534-198507000-00010.


    I. Pawłow, Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids, Discrete Contin. Dyn. Syst., 15 (2006), 1169-1191.doi: 10.3934/dcds.2006.15.1169.


    I. Pawłow and W. M. Zajączkowski, A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures, Commun. Pure Appl. Anal., 10 (2011), 1823-1847.doi: 10.3934/cpaa.2011.10.1823.


    T. V. Savina, A. A. Golovin, S. H. Davis, A. A. Nepomnyashchy and P. W. Voorhees, Faceting of a growing crystal surface by surface diffusion, Phys. Rev. E, 67 (2003), 021606.doi: 10.1103/PhysRevE.67.021606.


    G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, to appear, 2011.


    I. Singer-Loginova and H. M. Singer, The phase field technique for modeling multiphase materials, Rep. Prog. Phys., 71 (2008), 106501 (32 pp).doi: 10.1088/0034-4885/71/10/106501.


    V. A. Solonnikov, A priori estimates for solutions of second order parabolic equations, Trudy Mat. Inst. Steklov, 70 (1964), 133-212 (in Russian).


    V. A. Solonnikov, Boundary value problems for linear parabolic systems of differential equations of general type, Trudy Mat. Inst. Steklov, 83 (1965), 1-162 (in Russian).


    W. von Wahl, "The Equations of Navier-Stokes and Abstract Parabolic Equations," Braunschweig, 1985.

  • 加载中

Article Metrics

HTML views() PDF downloads(103) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint