doi: 10.3934/dcds.2021198
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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 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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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