May  2022, 42(5): 2453-2460. doi: 10.3934/dcds.2021198

A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials

SUSTech International Center for Mathematics, and Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China

Received  July 2020 Published  May 2022 Early access  January 2022

We introduce a regularization-free approach for the wellposedness of the classic Cahn-Hilliard equation with logarithmic potentials.

Citation: Dong Li. A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2453-2460. doi: 10.3934/dcds.2021198
References:
[1]

H. Abels and M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67 (2007), 3176-3193.  doi: 10.1016/j.na.2006.10.002.

[2]

P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[3]

P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differential Equations, 212 (2005), 235-277.  doi: 10.1016/j.jde.2004.07.003.

[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: 10.1063/1.1744102.

[5]

L. CherfilsA. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.  doi: 10.1007/s00032-011-0165-4.

[6]

M. Copetti and C. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65.  doi: 10.1007/BF01385847.

[7]

A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514.  doi: 10.1016/0362-546X(94)00205-V.

[8]

C. M. Elliott and S. Luckhaus, A generalized Diffusion Equation for Phase Separation of A Multi-Component Mixture with Interfacial Energy, SFB 256 Preprint No. 195, University of Bonn, 1991.

[9]

H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31.  doi: 10.1016/S0022-247X(02)00425-0.

[10]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interaction I. Macroscopic limits, J. Statist. Phys., 87 (1997), 37-61.  doi: 10.1007/BF02181479.

[11]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interaction II. Interface motion, SIAM J. Appl. Math., 58 (1998), 1707-1729.  doi: 10.1137/S0036139996313046.

[12]

N. KenmochiM. Niezgódka and I. Pawlow, Subdifferential operator approach to the Cahn-Hilliard equation with constraint, J. Differential Equations, 117 (1995), 320-356.  doi: 10.1006/jdeq.1995.1056.

[13]

D. Li, Effective maximum principles for spectral methods, Ann. Appl. Math., 37 (2021), 131-290.  doi: 10.4208/aam.OA-2021-0003.

[14]

D. Li, C. Quan and T. Tang, Stability and convergence analysis for the implicit-explicit discretization of the Cahn-Hilliard equation, to appear in Math. Comp., arXiv: 2008.03701.

[15]

D. Li and T. Tang, Stability of the semi-implicit method for the cahn-hilliard equation with logarithmic potentials, Ann. Appl. Math., 37 (2021), 31-60.  doi: 10.4208/aam.OA-2020-0003.

[16]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.

show all references

References:
[1]

H. Abels and M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67 (2007), 3176-3193.  doi: 10.1016/j.na.2006.10.002.

[2]

P. W. BatesP. C. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[3]

P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differential Equations, 212 (2005), 235-277.  doi: 10.1016/j.jde.2004.07.003.

[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: 10.1063/1.1744102.

[5]

L. CherfilsA. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.  doi: 10.1007/s00032-011-0165-4.

[6]

M. Copetti and C. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65.  doi: 10.1007/BF01385847.

[7]

A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514.  doi: 10.1016/0362-546X(94)00205-V.

[8]

C. M. Elliott and S. Luckhaus, A generalized Diffusion Equation for Phase Separation of A Multi-Component Mixture with Interfacial Energy, SFB 256 Preprint No. 195, University of Bonn, 1991.

[9]

H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31.  doi: 10.1016/S0022-247X(02)00425-0.

[10]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interaction I. Macroscopic limits, J. Statist. Phys., 87 (1997), 37-61.  doi: 10.1007/BF02181479.

[11]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interaction II. Interface motion, SIAM J. Appl. Math., 58 (1998), 1707-1729.  doi: 10.1137/S0036139996313046.

[12]

N. KenmochiM. Niezgódka and I. Pawlow, Subdifferential operator approach to the Cahn-Hilliard equation with constraint, J. Differential Equations, 117 (1995), 320-356.  doi: 10.1006/jdeq.1995.1056.

[13]

D. Li, Effective maximum principles for spectral methods, Ann. Appl. Math., 37 (2021), 131-290.  doi: 10.4208/aam.OA-2021-0003.

[14]

D. Li, C. Quan and T. Tang, Stability and convergence analysis for the implicit-explicit discretization of the Cahn-Hilliard equation, to appear in Math. Comp., arXiv: 2008.03701.

[15]

D. Li and T. Tang, Stability of the semi-implicit method for the cahn-hilliard equation with logarithmic potentials, Ann. Appl. Math., 37 (2021), 31-60.  doi: 10.4208/aam.OA-2020-0003.

[16]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.

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