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January  2013, 12(1): 461-480. doi: 10.3934/cpaa.2013.12.461

Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$

Sergey Zelik 1, and  Jon Pennant 2,

1. 

Department of Mathematics, University of Surrey, Guildford, GU2 7XH
2. 

University of Surrey, Guildford, GU2 7XH, China

Received  July 2011 Revised  May 2012 Published  September 2012

We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of arbitrary polynomial growth and for the case of sufficiently strong singular potentials. For these cases, the uniqueness and further regularity of the obtained solutions are proved as well. We discuss also the analogous problems for the case of the so-called Cahn-Hilliard-Oono equation where, in addition, the dissipativity of the associated solution semigroup is established.
Keywords: infinite-energy solutions., unbounded domains, Cahn-Hilliard equation.
Mathematics Subject Classification: 35B41, 35L05, 74K1.
Citation: Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure and Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461
References:
[1]

F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations, 83 (1990), 85-108. doi: 10.1016/0022-0396(90)90070-6.

[2]

A. Babin, Global attractors in PDE. Handbook of dynamical systems, Vol. 1B, 983-1085, Elsevier B. V., Amsterdam, 2006. doi: 10.1016/S1874-575X(06)80039-1.

[3]

A. V. Babin, M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992.

[4]

A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain, Proc. Royal. Soc. Edimburgh, 116A (1990), 221-243. doi: 10.1017/S0308210500031498.

[5]

A. Bonfoh, Finite-dimensional attractor for the viscious Cahn-Hilliard equation in an unbounded domain, Quarterly of Applied Mathematics, 64 (2006), 94-104.

[6]

J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts, Comm. Pure Appl. Math., 52 (1999), 839-871. doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I.

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.

[8]

L. Caffarelli and N. Muler, An $L^\infty$ bound for solutions of the Cahn-Hilliard equation, rch. Rational Mech. Anal., 133 (1995), 129-144. doi: 10.1007/BF00376814.

[9]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. doi: 10.1007/s00032-011-0165-4.

[10]

A. Debussche, A singular perturbation of the Cahn-Hilliard equation, Asymptotic Anal., 4 (1991), 161-185. doi: 10.3233/ASY-1991-4202.

[11]

T. Dlotko, M. Kania and C. Sun, Analysis of the viscous Cahn-Hilliard equation in $\R^N$, Journal Diff. Eqns., 252 (2012), 2771-2791. doi: 10.1016/j.jde.2011.08.052.

[12]

A. Eden and V. K. Kalantarov, 3D convective Cahn - Hilliard equation, Comm. Pure Appl. Anal., 6 (2007), 1075-1086. doi: 10.3934/cpaa.2007.6.1075.

[13]

A. Eden, V. Kalantarov and S. Zelik, Infinite-energy solutions for the Cahn-Hilliard equation in cylindrical domains,, submitted. \arXiv{1005.3424}, (). 

[14]

C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation. Mathematical models for phase change problems, Internat. Ser. Numer. Math., Birkhauser, Basel, 88 (1989), 35-73.

[15]

M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011.

[16]

M. Efendiev, H. Gajewski and S. Zelik, The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity, Adv. Differential Equations, 7 (2002), 1073-1100.

[17]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186.

[18]

J. Evans, V. Galaktionov and J. Williams, Blow-up and global asymptotics of the unstable Cahn-Hilliard equation with a homogeneous nonlinearity, SIAM Journal on Mathematical Analysis, 38 (2006), 64-102.

[19]

M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Comm. Partial Differential Equations, 34 (2009), 137-170. doi: 10.1080/03605300802608247.

[20]

M. Grasselli, Maurizio, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Equ., 9 (2009), 371-404. doi: 10.1007/s00028-009-0017-7.

[21]

M. Grasselli, H. Petzeltova and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Differential Equations, 239 (2007), 38-60. doi: 10.1016/j.jde.2007.05.003.

[22]

V. Kalantarov, Global behavior of the solutions of some fourth-order nonlinear equations. (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 163 (1987), Kraev. Zadachi Mat. Fiz. i Smezhn. Vopr. Teor. Funktsii 19, 66-75. doi: 10.1007/BF02208712.

[23]

T. Korvola, A. Kupiainen and J. Taskinen, Anomalous scaling for 3D Cahn-Hilliard Fronts, Comm. Pure Appl. Math., 58 (2005), 1077-1115.

[24]

A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360.

[25]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, 103-200, Handb. Differ. Equ., Elsevier North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[26]

A. Miranville and A. S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.

[27]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464.

[28]

A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation, Journal of Applied Analysis and Computation, 1 (2011), 523-536.

[29]

A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985.

[30]

A. Novick-Cohen, Blow up and growth in the directional solidification of dilute binary alloys, Appl. Anal., 47 (1992), 241-257. doi: 10.1080/00036819208840143.

[31]

Y. Oono and S. Puri, Computionally efficient modeling of ordering of quenched phases, Phys. Rev. Letters, 58 (1987), 836-839. doi: 10.1103/PhysRevLett.58.836.

[32]

E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems, Phys. D, 192 (2004), 279-307. doi: 10.1016/j.physd.2004.01.024.

[33]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.

[34]

J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincar Anal. Non Lineaire, 15 (1998), 459-492. doi: 10.1016/S0294-1449(98)80031-0.

[35]

S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains. Instability in models connected with fluid flows. II, Int. Math. Ser. Springer, New York, (N. Y.), 7 (2008), 255-327. doi: 10.1007/978-0-387-75219-8_6.

[36]

S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 49 (2007), 525-588. doi: 10.1017/S0017089507003849.

[37]

S. Zelik, Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains, J. Dynam. Differential Equations, 19 (2007), 1-74. doi: 10.1007/s10884-006-9007-4.

[38]

S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068.

show all references

References:
[1]

F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations, 83 (1990), 85-108. doi: 10.1016/0022-0396(90)90070-6.

[2]

A. Babin, Global attractors in PDE. Handbook of dynamical systems, Vol. 1B, 983-1085, Elsevier B. V., Amsterdam, 2006. doi: 10.1016/S1874-575X(06)80039-1.

[3]

A. V. Babin, M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992.

[4]

A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain, Proc. Royal. Soc. Edimburgh, 116A (1990), 221-243. doi: 10.1017/S0308210500031498.

[5]

A. Bonfoh, Finite-dimensional attractor for the viscious Cahn-Hilliard equation in an unbounded domain, Quarterly of Applied Mathematics, 64 (2006), 94-104.

[6]

J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts, Comm. Pure Appl. Math., 52 (1999), 839-871. doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I.

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.

[8]

L. Caffarelli and N. Muler, An $L^\infty$ bound for solutions of the Cahn-Hilliard equation, rch. Rational Mech. Anal., 133 (1995), 129-144. doi: 10.1007/BF00376814.

[9]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. doi: 10.1007/s00032-011-0165-4.

[10]

A. Debussche, A singular perturbation of the Cahn-Hilliard equation, Asymptotic Anal., 4 (1991), 161-185. doi: 10.3233/ASY-1991-4202.

[11]

T. Dlotko, M. Kania and C. Sun, Analysis of the viscous Cahn-Hilliard equation in $\R^N$, Journal Diff. Eqns., 252 (2012), 2771-2791. doi: 10.1016/j.jde.2011.08.052.

[12]

A. Eden and V. K. Kalantarov, 3D convective Cahn - Hilliard equation, Comm. Pure Appl. Anal., 6 (2007), 1075-1086. doi: 10.3934/cpaa.2007.6.1075.

[13]

A. Eden, V. Kalantarov and S. Zelik, Infinite-energy solutions for the Cahn-Hilliard equation in cylindrical domains,, submitted. \arXiv{1005.3424}, (). 

[14]

C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation. Mathematical models for phase change problems, Internat. Ser. Numer. Math., Birkhauser, Basel, 88 (1989), 35-73.

[15]

M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011.

[16]

M. Efendiev, H. Gajewski and S. Zelik, The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity, Adv. Differential Equations, 7 (2002), 1073-1100.

[17]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186.

[18]

J. Evans, V. Galaktionov and J. Williams, Blow-up and global asymptotics of the unstable Cahn-Hilliard equation with a homogeneous nonlinearity, SIAM Journal on Mathematical Analysis, 38 (2006), 64-102.

[19]

M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Comm. Partial Differential Equations, 34 (2009), 137-170. doi: 10.1080/03605300802608247.

[20]

M. Grasselli, Maurizio, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Equ., 9 (2009), 371-404. doi: 10.1007/s00028-009-0017-7.

[21]

M. Grasselli, H. Petzeltova and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Differential Equations, 239 (2007), 38-60. doi: 10.1016/j.jde.2007.05.003.

[22]

V. Kalantarov, Global behavior of the solutions of some fourth-order nonlinear equations. (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 163 (1987), Kraev. Zadachi Mat. Fiz. i Smezhn. Vopr. Teor. Funktsii 19, 66-75. doi: 10.1007/BF02208712.

[23]

T. Korvola, A. Kupiainen and J. Taskinen, Anomalous scaling for 3D Cahn-Hilliard Fronts, Comm. Pure Appl. Math., 58 (2005), 1077-1115.

[24]

A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360.

[25]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, 103-200, Handb. Differ. Equ., Elsevier North-Holland, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[26]

A. Miranville and A. S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.

[27]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464.

[28]

A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation, Journal of Applied Analysis and Computation, 1 (2011), 523-536.

[29]

A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985.

[30]

A. Novick-Cohen, Blow up and growth in the directional solidification of dilute binary alloys, Appl. Anal., 47 (1992), 241-257. doi: 10.1080/00036819208840143.

[31]

Y. Oono and S. Puri, Computionally efficient modeling of ordering of quenched phases, Phys. Rev. Letters, 58 (1987), 836-839. doi: 10.1103/PhysRevLett.58.836.

[32]

E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems, Phys. D, 192 (2004), 279-307. doi: 10.1016/j.physd.2004.01.024.

[33]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.

[34]

J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincar Anal. Non Lineaire, 15 (1998), 459-492. doi: 10.1016/S0294-1449(98)80031-0.

[35]

S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains. Instability in models connected with fluid flows. II, Int. Math. Ser. Springer, New York, (N. Y.), 7 (2008), 255-327. doi: 10.1007/978-0-387-75219-8_6.

[36]

S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 49 (2007), 525-588. doi: 10.1017/S0017089507003849.

[37]

S. Zelik, Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains, J. Dynam. Differential Equations, 19 (2007), 1-74. doi: 10.1007/s10884-006-9007-4.

[38]

S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068.

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