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Estimating area of inclusions in anisotropic plates from boundary data
On a class of sixth order viscous Cahn-Hilliard type equations
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 |
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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