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 & Continuous Dynamical Systems - A, 2016, 36 (10) : 5387-5400. doi: 10.3934/dcds.2016037
References:
[1]

R. A. Adams, Sobolev Space,, Academic press, (1975).   Google Scholar

[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.  doi: 10.1007/BF00375025.  Google Scholar

[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.  doi: 10.1137/0719018.  Google Scholar

[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.  doi: 10.1137/0723049.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  doi: 10.1090/S0025-5718-1992-1122067-1.  Google Scholar

[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.  doi: 10.1090/S0025-5718-06-01915-6.  Google Scholar

[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.  doi: 10.1007/s00211-004-0546-5.  Google Scholar

[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).  doi: 10.1137/130928662.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1002/num.20328.  Google Scholar

[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.  doi: 10.1016/j.apnum.2006.07.026.  Google Scholar

[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.  Google Scholar

[16]

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

show all references

References:
[1]

R. A. Adams, Sobolev Space,, Academic press, (1975).   Google Scholar

[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.  doi: 10.1007/BF00375025.  Google Scholar

[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.  doi: 10.1137/0719018.  Google Scholar

[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.  doi: 10.1137/0723049.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  doi: 10.1090/S0025-5718-1992-1122067-1.  Google Scholar

[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.  doi: 10.1090/S0025-5718-06-01915-6.  Google Scholar

[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.  doi: 10.1007/s00211-004-0546-5.  Google Scholar

[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).  doi: 10.1137/130928662.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1002/num.20328.  Google Scholar

[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.  doi: 10.1016/j.apnum.2006.07.026.  Google Scholar

[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.  Google Scholar

[16]

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

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