October  2016, 36(10): 5387-5400. doi: 10.3934/dcds.2016037

On uniform in time $H^2$-regularity of the solution for the 2D Cahn-Hilliard equation

1. 

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

2. 

Center for Computational Geosciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049

Received  October 2015 Revised  March 2016 Published  July 2016

In this article, we provide the uniform $H^2$-regularity results with respect to $t$ of the solution and its time derivatives for the 2D Cahn-Hilliard equation. Based on sharp a priori estimates for the solution of problem under the assumption on the initial value, we show that the $H^2$-regularity of the solution and its first and second order time derivatives only depend on $\epsilon^{-1}$.
Citation: Xinlong Feng, Yinnian He. On uniform in time $H^2$-regularity of the solution for the 2D Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5387-5400. doi: 10.3934/dcds.2016037
References:
[1]

R. A. Adams, Sobolev Space, Academic press, New York, 1975.

[2]

N. D. Alikakos, P. W. Bates and X. F. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205. doi: 10.1007/BF00375025.

[3]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311. doi: 10.1137/0719018.

[4]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, part II: Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal., 23 (1986), 750-777. doi: 10.1137/0723049.

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, I: Interfacial free energy, J. Chem. Phys., 28 (1958), 258-268.

[6]

Q. Du and R. A. Nicolaides, Numerical analysia of a continum model of phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322. doi: 10.1137/0728069.

[7]

C. M. Elliott and S. M. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.

[8]

C. M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comp., 58 (1992), 603-630. doi: 10.1090/S0025-5718-1992-1122067-1.

[9]

X. B. Feng, Y. N. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase fluids, Math. Comp., 76 (2007), 539-571. doi: 10.1090/S0025-5718-06-01915-6.

[10]

X. B. Feng and A. Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math., 99 (2004), 47-84. doi: 10.1007/s00211-004-0546-5.

[11]

X. L. Feng, T. Tang and J. Yang, Long time numerical simulations for phase-field problems using p-adaptive spectral deferred correction methods, SIAM J. Sci. Comput., 37 (2015), A271-A294. doi: 10.1137/130928662.

[12]

X. L. Feng, T. Tang and J. Yang, Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models, East Asian J. Appl. Math., 3 (2013), 59-80.

[13]

Y. N. He and Y. X. Liu, Stability and convergence of the spectral Galerkin method for the Cahn-Hilliard equation, Numer. Meth. Part Differ. Equ., 24 (2008), 1485-1500. doi: 10.1002/num.20328.

[14]

Y. N. He, Y. X. Liu and T. Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007), 616-628. doi: 10.1016/j.apnum.2006.07.026.

[15]

J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Volume 41 of Springer Series in Computational Mathematics. Springer, 2011. doi: 10.1007/978-3-540-71041-7.

[16]

J. Shen and X. F. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discret. Contin. Dyn. Syst., 28 (2010), 1669-1691. doi: 10.3934/dcds.2010.28.1669.

show all references

References:
[1]

R. A. Adams, Sobolev Space, Academic press, New York, 1975.

[2]

N. D. Alikakos, P. W. Bates and X. F. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205. doi: 10.1007/BF00375025.

[3]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311. doi: 10.1137/0719018.

[4]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, part II: Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal., 23 (1986), 750-777. doi: 10.1137/0723049.

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, I: Interfacial free energy, J. Chem. Phys., 28 (1958), 258-268.

[6]

Q. Du and R. A. Nicolaides, Numerical analysia of a continum model of phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322. doi: 10.1137/0728069.

[7]

C. M. Elliott and S. M. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.

[8]

C. M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comp., 58 (1992), 603-630. doi: 10.1090/S0025-5718-1992-1122067-1.

[9]

X. B. Feng, Y. N. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase fluids, Math. Comp., 76 (2007), 539-571. doi: 10.1090/S0025-5718-06-01915-6.

[10]

X. B. Feng and A. Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math., 99 (2004), 47-84. doi: 10.1007/s00211-004-0546-5.

[11]

X. L. Feng, T. Tang and J. Yang, Long time numerical simulations for phase-field problems using p-adaptive spectral deferred correction methods, SIAM J. Sci. Comput., 37 (2015), A271-A294. doi: 10.1137/130928662.

[12]

X. L. Feng, T. Tang and J. Yang, Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models, East Asian J. Appl. Math., 3 (2013), 59-80.

[13]

Y. N. He and Y. X. Liu, Stability and convergence of the spectral Galerkin method for the Cahn-Hilliard equation, Numer. Meth. Part Differ. Equ., 24 (2008), 1485-1500. doi: 10.1002/num.20328.

[14]

Y. N. He, Y. X. Liu and T. Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007), 616-628. doi: 10.1016/j.apnum.2006.07.026.

[15]

J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Volume 41 of Springer Series in Computational Mathematics. Springer, 2011. doi: 10.1007/978-3-540-71041-7.

[16]

J. Shen and X. F. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discret. Contin. Dyn. Syst., 28 (2010), 1669-1691. doi: 10.3934/dcds.2010.28.1669.

[1]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127

[2]

Kelong Cheng, Cheng Wang, Steven M. Wise, Zixia Yuan. Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2211-2229. doi: 10.3934/dcdss.2020186

[3]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31

[4]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163

[5]

Matthieu Brachet, Philippe Parnaudeau, Morgan Pierre. Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022110

[6]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

[7]

Andreas C. Aristotelous, Ohannes Karakashian, Steven M. Wise. A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2211-2238. doi: 10.3934/dcdsb.2013.18.2211

[8]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207

[9]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013

[10]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033

[11]

Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 763-782. doi: 10.3934/cpaa.2020289

[12]

Fausto Cavalli, Giovanni Naldi. A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation. Kinetic and Related Models, 2010, 3 (1) : 123-142. doi: 10.3934/krm.2010.3.123

[13]

Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625

[14]

Maurizio Grasselli, Nicolas Lecoq, Morgan Pierre. A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term. Conference Publications, 2011, 2011 (Special) : 543-552. doi: 10.3934/proc.2011.2011.543

[15]

Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations and Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391

[16]

L. Chupin. Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 45-68. doi: 10.3934/dcdsb.2003.3.45

[17]

Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303

[18]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037

[19]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275

[20]

S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (149)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]