March  2020, 40(3): 1847-1878. doi: 10.3934/dcds.2020096

Convergence to equilibrium for a bulk–surface Allen–Cahn system coupled through a nonlinear Robin boundary condition

1. 

Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China

2. 

School of Mathematical Sciences, Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Handan Road 220, Shanghai 200433, China

* Corresponding author: Hao Wu

Received  July 2019 Revised  October 2019 Published  December 2019

Fund Project: K. F. Lam is supported by a Direct Grant of CUHK (project 4053288), H. Wu is supported by NNSFC grant No. 11631011 and the Shanghai Center for Mathematical Sciences at Fudan University.

We consider a coupled bulk–surface Allen–Cahn system affixed with a Robin-type boundary condition between the bulk and surface variables. This system can also be viewed as a relaxation to a bulk–surface Allen–Cahn system with non-trivial transmission conditions. Assuming that the nonlinearities are real analytic, we prove the convergence of every global strong solution to a single equilibrium as time tends to infinity. Furthermore, we obtain an estimate on the rate of convergence. The proof relies on an extended Łojasiewicz–Simon type inequality for the bulk–surface coupled system. Compared with previous works, new difficulties arise as in our system the surface variable is no longer the trace of the bulk variable, but now they are coupled through a nonlinear Robin boundary condition.

Citation: Kei Fong Lam, Hao Wu. Convergence to equilibrium for a bulk–surface Allen–Cahn system coupled through a nonlinear Robin boundary condition. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1847-1878. doi: 10.3934/dcds.2020096
References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Met., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.

[2]

I. Babuška, The finite element method with penalty, Math. Comput., 27 (1973), 221-228.  doi: 10.1090/S0025-5718-1973-0351118-5.

[3]

J. W. Barrett and C. M. Elliott, Finite element approximation of the Dirichlet problem using the boundary penalty method, Numer. Math., 49 (1986), 343-366.  doi: 10.1007/BF01389536.

[4]

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: 10.1002/9781118788295.ch4.

[5]

L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27.  doi: 10.1016/j.na.2012.11.010.

[6]

C. CavaterraC. G. GalM. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399.  doi: 10.1016/j.na.2009.11.002.

[7]

C. CavaterraM. Grasselli and H. Wu, Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions, Commun. Pure Appl. Anal., 13 (2014), 1855-1890.  doi: 10.3934/cpaa.2014.13.1855.

[8]

X. F. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.

[9]

R. ChillE. Făsangová and J. Prüss, Convergence to steady state of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.  doi: 10.1002/mana.200410431.

[10]

P. Colli and T. Fukao, The Allen-Cahn equation with dynamic boundary conditions and mass constraints, Math. Models Appl. Sci., 38 (2015), 3950-3967.  doi: 10.1002/mma.3329.

[11]

P. Colli and T. Fukao, Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials, Nonlinear Anal., 127 (2015), 413-433.  doi: 10.1016/j.na.2015.07.011.

[12]

P. ColliT. Fukao and K. F. Lam, On a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition, Nonlinear Anal., 184 (2019), 116-147.  doi: 10.1016/j.na.2018.10.018.

[13]

P. Colli, T. Fukao and H. Wu, On a transmission problem for equation and dynamic boundary condition of Cahn-Hilliard type with nonsmooth potentials, Math. Nachr., accepted for publication, 2019.

[14]

P. ColliG. GilardiR. Nakayashiki and K. Shirakawa, A class of quasi-linear Allen-Cahn type equations with dynamic boundary conditions, Nonlinear Anal., 158 (2017), 32-59.  doi: 10.1016/j.na.2017.03.020.

[15]

P. ColliG. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994.  doi: 10.1016/j.jmaa.2014.05.008.

[16]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.  doi: 10.1090/S0002-9947-1995-1672406-7.

[17]

C. M. ElliottT. Ranner and C. Venkataraman, Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics, SIAM J. Math. Anal., 49 (2017), 360-397.  doi: 10.1137/15M1050811.

[18]

A. FaviniG. R. GoldsteinJ. A. Goldstein and S. Romanelli, The heat equation with nonlinear general Wentzell boundary condition, Adv. Differential Equations, 11 (2006), 481-510. 

[19]

C. G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls, Math. Meth. Appl. Sci., 29 (2006), 2009-2036.  doi: 10.1002/mma.757.

[20]

C. G. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009.

[21]

C. G. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 689-710.  doi: 10.3934/cpaa.2009.8.689.

[22]

C. G. Gal and H. Wu, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst., 22 (2008), 1041-1063.  doi: 10.3934/dcds.2008.22.1041.

[23]

H. Garcke and P. Knopf, Weak solutions of the Cahn-Hilliard system with dynamic boundary conditions: A gradient flow approach, preprint, arXiv: 1810.09817.

[24]

G. R. GoldsteinA. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non permeable walls, Physica D, 240 (2011), 754-766.  doi: 10.1016/j.physd.2010.12.007.

[25]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[26]

A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var., 9 (1999), 95-124.  doi: 10.1007/s005260050133.

[27]

A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal., 26 (2001), 21-36. 

[28]

T. Illmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom., 38 (1993), 417-461.  doi: 10.4310/jdg/1214454300.

[29]

M. A. Jendoubi, A simple unified approach to some convergence theorem of L. Simon, J. Funct. Anal., 153 (1998), 187-202.  doi: 10.1006/jfan.1997.3174.

[30]

R. KenzlerF. EurichP. MaassB. RinnJ. SchroppE. Bohl and W. Dieterich, Phase separation in confined geometries: solving the Cahn-Hilliard equation with generic boundary conditions, Comput. Phys. Commun., 133 (2001), 139-157.  doi: 10.1016/S0010-4655(00)00159-4.

[31]

M. Kleman and O. D. Lavrentovich, Topological point defects in nematic liquid crystals, Phil. Mag., 86 (2004), 4117-4137.  doi: 10.1080/14786430600593016.

[32]

P. Knopf and K. F. Lam, Convergence of a Robin boundary approximation for a Cahn-Hilliard system with dynamic boundary conditions, preprint, arXiv: 1908.06124.

[33]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

[34]

C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.

[35]

C. Liu and H. Wu, An energetic variational approach for the Cahn-Hilliard equation with dynamic boundary condition: Model derivation and mathematical analysis, Arch. Rational Mech. Anal., 233 (2019), 167-247.  doi: 10.1007/s00205-019-01356-x.

[36]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst., 28 (2010), 275-310.  doi: 10.3934/dcds.2010.28.275.

[37]

T. Motoda, Time periodic solutions of Cahn-Hilliard systems with dynamic boundary conditions, AIMS Math., 3 (2018), 263-287. 

[38]

T. Z. QianX. P. Wang and P. Sheng, A variational approach to moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333-360.  doi: 10.1017/S0022112006001935.

[39]

R. Racke and S. M. Zheng, The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Differential Equations, 8 (2003), 83-110. 

[40]

L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525-571.  doi: 10.2307/2006981.

[41]

J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary conditions, Nonlinear Anal., 72 (2010), 3028-3048.  doi: 10.1016/j.na.2009.11.043.

[42]

K. Taira, Semigroups, Boundary Value Problems, and Markov Process, Springer Monographs in Mathematics, Springer-Verlag Berlin Heidelberg, 2004. doi: 10.1007/978-3-662-09857-8.

[43]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[44]

H. Wu, Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition, Asymptot. Anal., 54 (2007), 71-92. 

[45]

H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamical boundary condition, Adv. Math. Sci. Appl., 17 (2007), 67-88. 

[46]

H. Wu and S. M. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions, J. Differential Equations, 204 (2004), 511-531.  doi: 10.1016/j.jde.2004.05.004.

[47]

E. Zeidler, Nonlinear Functional Analysis and its Applications I, Fixed-Point Theorems, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.

[48]

S. M. Zheng, Nonlinear Evolution Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133. Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.

show all references

References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Met., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.

[2]

I. Babuška, The finite element method with penalty, Math. Comput., 27 (1973), 221-228.  doi: 10.1090/S0025-5718-1973-0351118-5.

[3]

J. W. Barrett and C. M. Elliott, Finite element approximation of the Dirichlet problem using the boundary penalty method, Numer. Math., 49 (1986), 343-366.  doi: 10.1007/BF01389536.

[4]

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: 10.1002/9781118788295.ch4.

[5]

L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27.  doi: 10.1016/j.na.2012.11.010.

[6]

C. CavaterraC. G. GalM. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399.  doi: 10.1016/j.na.2009.11.002.

[7]

C. CavaterraM. Grasselli and H. Wu, Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions, Commun. Pure Appl. Anal., 13 (2014), 1855-1890.  doi: 10.3934/cpaa.2014.13.1855.

[8]

X. F. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.

[9]

R. ChillE. Făsangová and J. Prüss, Convergence to steady state of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.  doi: 10.1002/mana.200410431.

[10]

P. Colli and T. Fukao, The Allen-Cahn equation with dynamic boundary conditions and mass constraints, Math. Models Appl. Sci., 38 (2015), 3950-3967.  doi: 10.1002/mma.3329.

[11]

P. Colli and T. Fukao, Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials, Nonlinear Anal., 127 (2015), 413-433.  doi: 10.1016/j.na.2015.07.011.

[12]

P. ColliT. Fukao and K. F. Lam, On a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition, Nonlinear Anal., 184 (2019), 116-147.  doi: 10.1016/j.na.2018.10.018.

[13]

P. Colli, T. Fukao and H. Wu, On a transmission problem for equation and dynamic boundary condition of Cahn-Hilliard type with nonsmooth potentials, Math. Nachr., accepted for publication, 2019.

[14]

P. ColliG. GilardiR. Nakayashiki and K. Shirakawa, A class of quasi-linear Allen-Cahn type equations with dynamic boundary conditions, Nonlinear Anal., 158 (2017), 32-59.  doi: 10.1016/j.na.2017.03.020.

[15]

P. ColliG. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994.  doi: 10.1016/j.jmaa.2014.05.008.

[16]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.  doi: 10.1090/S0002-9947-1995-1672406-7.

[17]

C. M. ElliottT. Ranner and C. Venkataraman, Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics, SIAM J. Math. Anal., 49 (2017), 360-397.  doi: 10.1137/15M1050811.

[18]

A. FaviniG. R. GoldsteinJ. A. Goldstein and S. Romanelli, The heat equation with nonlinear general Wentzell boundary condition, Adv. Differential Equations, 11 (2006), 481-510. 

[19]

C. G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls, Math. Meth. Appl. Sci., 29 (2006), 2009-2036.  doi: 10.1002/mma.757.

[20]

C. G. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009.

[21]

C. G. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 689-710.  doi: 10.3934/cpaa.2009.8.689.

[22]

C. G. Gal and H. Wu, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst., 22 (2008), 1041-1063.  doi: 10.3934/dcds.2008.22.1041.

[23]

H. Garcke and P. Knopf, Weak solutions of the Cahn-Hilliard system with dynamic boundary conditions: A gradient flow approach, preprint, arXiv: 1810.09817.

[24]

G. R. GoldsteinA. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non permeable walls, Physica D, 240 (2011), 754-766.  doi: 10.1016/j.physd.2010.12.007.

[25]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[26]

A. Haraux and M. A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var., 9 (1999), 95-124.  doi: 10.1007/s005260050133.

[27]

A. Haraux and M. A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptot. Anal., 26 (2001), 21-36. 

[28]

T. Illmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom., 38 (1993), 417-461.  doi: 10.4310/jdg/1214454300.

[29]

M. A. Jendoubi, A simple unified approach to some convergence theorem of L. Simon, J. Funct. Anal., 153 (1998), 187-202.  doi: 10.1006/jfan.1997.3174.

[30]

R. KenzlerF. EurichP. MaassB. RinnJ. SchroppE. Bohl and W. Dieterich, Phase separation in confined geometries: solving the Cahn-Hilliard equation with generic boundary conditions, Comput. Phys. Commun., 133 (2001), 139-157.  doi: 10.1016/S0010-4655(00)00159-4.

[31]

M. Kleman and O. D. Lavrentovich, Topological point defects in nematic liquid crystals, Phil. Mag., 86 (2004), 4117-4137.  doi: 10.1080/14786430600593016.

[32]

P. Knopf and K. F. Lam, Convergence of a Robin boundary approximation for a Cahn-Hilliard system with dynamic boundary conditions, preprint, arXiv: 1908.06124.

[33]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

[34]

C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.

[35]

C. Liu and H. Wu, An energetic variational approach for the Cahn-Hilliard equation with dynamic boundary condition: Model derivation and mathematical analysis, Arch. Rational Mech. Anal., 233 (2019), 167-247.  doi: 10.1007/s00205-019-01356-x.

[36]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst., 28 (2010), 275-310.  doi: 10.3934/dcds.2010.28.275.

[37]

T. Motoda, Time periodic solutions of Cahn-Hilliard systems with dynamic boundary conditions, AIMS Math., 3 (2018), 263-287. 

[38]

T. Z. QianX. P. Wang and P. Sheng, A variational approach to moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333-360.  doi: 10.1017/S0022112006001935.

[39]

R. Racke and S. M. Zheng, The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Differential Equations, 8 (2003), 83-110. 

[40]

L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math., 118 (1983), 525-571.  doi: 10.2307/2006981.

[41]

J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary conditions, Nonlinear Anal., 72 (2010), 3028-3048.  doi: 10.1016/j.na.2009.11.043.

[42]

K. Taira, Semigroups, Boundary Value Problems, and Markov Process, Springer Monographs in Mathematics, Springer-Verlag Berlin Heidelberg, 2004. doi: 10.1007/978-3-662-09857-8.

[43]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[44]

H. Wu, Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition, Asymptot. Anal., 54 (2007), 71-92. 

[45]

H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamical boundary condition, Adv. Math. Sci. Appl., 17 (2007), 67-88. 

[46]

H. Wu and S. M. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions, J. Differential Equations, 204 (2004), 511-531.  doi: 10.1016/j.jde.2004.05.004.

[47]

E. Zeidler, Nonlinear Functional Analysis and its Applications I, Fixed-Point Theorems, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.

[48]

S. M. Zheng, Nonlinear Evolution Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133. Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.

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