# American Institute of Mathematical Sciences

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.

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##### 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.
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