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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 $\mathbb{R}^3$

 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.
Citation: Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & 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. doi: 10.1016/0022-0396(90)90070-6. Google Scholar [2] A. Babin, Global attractors in PDE. Handbook of dynamical systems,, Vol. 1B, (2006), 983. doi: 10.1016/S1874-575X(06)80039-1. Google Scholar [3] A. V. Babin, M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, (1992). Google Scholar [4] A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain,, Proc. Royal. Soc. Edimburgh, 116A (1990), 221. doi: 10.1017/S0308210500031498. Google Scholar [5] A. Bonfoh, Finite-dimensional attractor for the viscious Cahn-Hilliard equation in an unbounded domain,, Quarterly of Applied Mathematics, 64 (2006), 94. Google Scholar [6] J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts,, Comm. Pure Appl. Math., 52 (1999), 839. doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I. Google Scholar [7] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. Google Scholar [8] L. Caffarelli and N. Muler, An $L^\infty$ bound for solutions of the Cahn-Hilliard equation,, rch. Rational Mech. Anal., 133 (1995), 129. doi: 10.1007/BF00376814. Google Scholar [9] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. doi: 10.1007/s00032-011-0165-4. Google Scholar [10] A. Debussche, A singular perturbation of the Cahn-Hilliard equation,, Asymptotic Anal., 4 (1991), 161. doi: 10.3233/ASY-1991-4202. Google Scholar [11] T. Dlotko, M. Kania and C. Sun, Analysis of the viscous Cahn-Hilliard equation in $\R^N$,, Journal Diff. Eqns., 252 (2012), 2771. doi: 10.1016/j.jde.2011.08.052. Google Scholar [12] A. Eden and V. K. Kalantarov, 3D convective Cahn - Hilliard equation,, Comm. Pure Appl. Anal., 6 (2007), 1075. doi: 10.3934/cpaa.2007.6.1075. Google Scholar [13] A. Eden, V. Kalantarov and S. Zelik, Infinite-energy solutions for the Cahn-Hilliard equation in cylindrical domains,, submitted. \arXiv{1005.3424}, (). Google Scholar [14] C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation. Mathematical models for phase change problems,, Internat. Ser. Numer. Math., 88 (1989), 35. Google Scholar [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. doi: 10.1002/cpa.1011. Google Scholar [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. Google Scholar [17] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system,, Math. Nachr., 272 (2004), 11. doi: 10.1002/mana.200310186. Google Scholar [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. Google Scholar [19] M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, Comm. Partial Differential Equations, 34 (2009), 137. doi: 10.1080/03605300802608247. Google Scholar [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. doi: 10.1007/s00028-009-0017-7. Google Scholar [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. doi: 10.1016/j.jde.2007.05.003. Google Scholar [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), 66. doi: 10.1007/BF02208712. Google Scholar [23] T. Korvola, A. Kupiainen and J. Taskinen, Anomalous scaling for 3D Cahn-Hilliard Fronts,, Comm. Pure Appl. Math., 58 (2005), 1077. Google Scholar [24] A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations,, Hokkaido Math. J., 38 (2009), 315. Google Scholar [25] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [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. doi: 10.1002/mma.590. Google Scholar [27] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials,, Math. Methods Appl. Sci., 27 (2004), 545. doi: 10.1002/mma.464. Google Scholar [28] A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation,, Journal of Applied Analysis and Computation, 1 (2011), 523. Google Scholar [29] A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965. Google Scholar [30] A. Novick-Cohen, Blow up and growth in the directional solidification of dilute binary alloys,, Appl. Anal., 47 (1992), 241. doi: 10.1080/00036819208840143. Google Scholar [31] Y. Oono and S. Puri, Computionally efficient modeling of ordering of quenched phases,, Phys. Rev. Letters, 58 (1987), 836. doi: 10.1103/PhysRevLett.58.836. Google Scholar [32] E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems,, Phys. D, 192 (2004), 279. doi: 10.1016/j.physd.2004.01.024. Google Scholar [33] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Second edition,, Applied Mathematical Sciences, 68 (1997). Google Scholar [34] J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation,, Ann. Inst. H. Poincar Anal. Non Lineaire, 15 (1998), 459. doi: 10.1016/S0294-1449(98)80031-0. Google Scholar [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, 7 (2008), 255. doi: 10.1007/978-0-387-75219-8_6. Google Scholar [36] S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip,, Glasg. Math. J., 49 (2007), 525. doi: 10.1017/S0017089507003849. Google Scholar [37] S. Zelik, Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains,, J. Dynam. Differential Equations, 19 (2007), 1. doi: 10.1007/s10884-006-9007-4. Google Scholar [38] S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity,, Comm. Pure Appl. Math., 56 (2003), 584. doi: 10.1002/cpa.10068. Google Scholar

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. doi: 10.1016/0022-0396(90)90070-6. Google Scholar [2] A. Babin, Global attractors in PDE. Handbook of dynamical systems,, Vol. 1B, (2006), 983. doi: 10.1016/S1874-575X(06)80039-1. Google Scholar [3] A. V. Babin, M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, (1992). Google Scholar [4] A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain,, Proc. Royal. Soc. Edimburgh, 116A (1990), 221. doi: 10.1017/S0308210500031498. Google Scholar [5] A. Bonfoh, Finite-dimensional attractor for the viscious Cahn-Hilliard equation in an unbounded domain,, Quarterly of Applied Mathematics, 64 (2006), 94. Google Scholar [6] J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts,, Comm. Pure Appl. Math., 52 (1999), 839. doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I. Google Scholar [7] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. Google Scholar [8] L. Caffarelli and N. Muler, An $L^\infty$ bound for solutions of the Cahn-Hilliard equation,, rch. Rational Mech. Anal., 133 (1995), 129. doi: 10.1007/BF00376814. Google Scholar [9] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. doi: 10.1007/s00032-011-0165-4. Google Scholar [10] A. Debussche, A singular perturbation of the Cahn-Hilliard equation,, Asymptotic Anal., 4 (1991), 161. doi: 10.3233/ASY-1991-4202. Google Scholar [11] T. Dlotko, M. Kania and C. Sun, Analysis of the viscous Cahn-Hilliard equation in $\R^N$,, Journal Diff. Eqns., 252 (2012), 2771. doi: 10.1016/j.jde.2011.08.052. Google Scholar [12] A. Eden and V. K. Kalantarov, 3D convective Cahn - Hilliard equation,, Comm. Pure Appl. Anal., 6 (2007), 1075. doi: 10.3934/cpaa.2007.6.1075. Google Scholar [13] A. Eden, V. Kalantarov and S. Zelik, Infinite-energy solutions for the Cahn-Hilliard equation in cylindrical domains,, submitted. \arXiv{1005.3424}, (). Google Scholar [14] C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation. Mathematical models for phase change problems,, Internat. Ser. Numer. Math., 88 (1989), 35. Google Scholar [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. doi: 10.1002/cpa.1011. Google Scholar [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. Google Scholar [17] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system,, Math. Nachr., 272 (2004), 11. doi: 10.1002/mana.200310186. Google Scholar [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. Google Scholar [19] M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, Comm. Partial Differential Equations, 34 (2009), 137. doi: 10.1080/03605300802608247. Google Scholar [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. doi: 10.1007/s00028-009-0017-7. Google Scholar [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. doi: 10.1016/j.jde.2007.05.003. Google Scholar [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), 66. doi: 10.1007/BF02208712. Google Scholar [23] T. Korvola, A. Kupiainen and J. Taskinen, Anomalous scaling for 3D Cahn-Hilliard Fronts,, Comm. Pure Appl. Math., 58 (2005), 1077. Google Scholar [24] A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations,, Hokkaido Math. J., 38 (2009), 315. Google Scholar [25] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [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. doi: 10.1002/mma.590. Google Scholar [27] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials,, Math. Methods Appl. Sci., 27 (2004), 545. doi: 10.1002/mma.464. Google Scholar [28] A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation,, Journal of Applied Analysis and Computation, 1 (2011), 523. Google Scholar [29] A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965. Google Scholar [30] A. Novick-Cohen, Blow up and growth in the directional solidification of dilute binary alloys,, Appl. Anal., 47 (1992), 241. doi: 10.1080/00036819208840143. Google Scholar [31] Y. Oono and S. Puri, Computionally efficient modeling of ordering of quenched phases,, Phys. Rev. Letters, 58 (1987), 836. doi: 10.1103/PhysRevLett.58.836. Google Scholar [32] E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems,, Phys. D, 192 (2004), 279. doi: 10.1016/j.physd.2004.01.024. Google Scholar [33] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Second edition,, Applied Mathematical Sciences, 68 (1997). Google Scholar [34] J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation,, Ann. Inst. H. Poincar Anal. Non Lineaire, 15 (1998), 459. doi: 10.1016/S0294-1449(98)80031-0. Google Scholar [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, 7 (2008), 255. doi: 10.1007/978-0-387-75219-8_6. Google Scholar [36] S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip,, Glasg. Math. J., 49 (2007), 525. doi: 10.1017/S0017089507003849. Google Scholar [37] S. Zelik, Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains,, J. Dynam. Differential Equations, 19 (2007), 1. doi: 10.1007/s10884-006-9007-4. Google Scholar [38] S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity,, Comm. Pure Appl. Math., 56 (2003), 584. doi: 10.1002/cpa.10068. Google Scholar
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