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

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April 2013, 6(2): 517-546. doi: 10.3934/dcdss.2013.6.517

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

Irena Pawłow ^1, and Wojciech M. Zajączkowski ^2,

1.	System Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw
2.	Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw

Received July 2011 Revised February 2012 Published November 2012

Abstract
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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.

Keywords: existence of global regular solutions, Sixth order viscous Cahn-Hilliard type equation, oil-water-surfactant mixtures..

Mathematics Subject Classification: Primary: 35K60; Secondary: 35Q72, 35L20.

Citation: Irena Pawłow, Wojciech M. Zajączkowski. On a class of sixth order viscous Cahn-Hilliard type equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 517-546. doi: 10.3934/dcdss.2013.6.517

References:

[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.
[2]	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.
[3]	O. V. Besov, V. P. Il'in and S. M. Nikolskij, "Integral Representation of Functions and Theorems of Imbeddings," Nauka, Moscow, 1975 (in Russian).
[4]	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.
[5]	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.
[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.
[7]	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.
[8]	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.
[9]	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.
[10]	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.
[11]	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.
[12]	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.
[13]	M. D. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation, to appear, 2011.
[14]	M. D. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation, to appear, 2011.
[15]	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.
[16]	I. Müller, "Thermodynamics," Pitman, London, 1985. doi: 10.1097/00006534-198507000-00010.
[17]	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.
[18]	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.
[19]	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.
[20]	G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, to appear, 2011.
[21]	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.
[22]	V. A. Solonnikov, A priori estimates for solutions of second order parabolic equations, Trudy Mat. Inst. Steklov, 70 (1964), 133-212 (in Russian).
[23]	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).
[24]	W. von Wahl, "The Equations of Navier-Stokes and Abstract Parabolic Equations," Braunschweig, 1985.

show all references

References:

[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.
[2]	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.
[3]	O. V. Besov, V. P. Il'in and S. M. Nikolskij, "Integral Representation of Functions and Theorems of Imbeddings," Nauka, Moscow, 1975 (in Russian).
[4]	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.
[5]	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.
[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.
[7]	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.
[8]	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.
[9]	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.
[10]	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.
[11]	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.
[12]	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.
[13]	M. D. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation, to appear, 2011.
[14]	M. D. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation, to appear, 2011.
[15]	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.
[16]	I. Müller, "Thermodynamics," Pitman, London, 1985. doi: 10.1097/00006534-198507000-00010.
[17]	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.
[18]	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.
[19]	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.
[20]	G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, to appear, 2011.
[21]	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.
[22]	V. A. Solonnikov, A priori estimates for solutions of second order parabolic equations, Trudy Mat. Inst. Steklov, 70 (1964), 133-212 (in Russian).
[23]	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).
[24]	W. von Wahl, "The Equations of Navier-Stokes and Abstract Parabolic Equations," Braunschweig, 1985.

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