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Shadowing as a structural property of the space of dynamical systems
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 |
We introduce a regularization-free approach for the wellposedness of the classic Cahn-Hilliard equation with logarithmic potentials.
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. Bates, P. C. Fife, X. 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. Cherfils, A. 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. Kenmochi, M. 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. Bates, P. C. Fife, X. 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. Cherfils, A. 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. Kenmochi, M. 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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