Numerical analysis for a nonlocal Allen-Cahn equation

International Journal of Numerical Analysis and Modeling

Volumn 6, Issue 1, 2009, Pages 33-49

  Bates, Peter W ^a   Brown, Sarah ^b   Han, Jianlong ^b  

^a MICHIGAN STATE UNIVERSITY   (United States)

^b SOUTHERN UTAH UNIVERSITY   (United States)

Author keywords

Finite difference scheme; Long range interaction

Indexed keywords

EID: 62749183716     PISSN: 17055105     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (18)

1. G. Barles and P. Souganidis, A new approach to front propagation problems: Theory and applications, Arch. Rational Mech. Anal. 141 (1998), 237-296.
2. P. W. Bates, On some nonlocal evolution equations arising from materials science Fields Inst. Communications, Vol 48(2006), 13-52.
3. P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solution in higher space dimensions, J. Stat. Phys. 95 (1999), 1119-1139.
4. P. W. Bates, X.Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice SIAM J. MATH. ANAL Vol. 35, No. 2 (2003), 520-546.
5. P. Bates, P. Fife, X. Ren, and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal. 138(1997), 105-136.
6. P. Bates, J. Han, and G. Zhao, On a nonlocal phase-field system, Nonlinear Analysis 64 (2006), No. 10, 2251-2278.
7. L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math.43 (1990), 983-997.
8. J. G. Carr, M. E. Gurtin, M. Slemrod, Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal. 86 (1984) 317-351.
9. X. Chen, Generation and propagation of interface for reaction-diffusion equations, J. Diff. Equations 96 (1992), 116-141.
10. E. Feireisl, F. I. Roch, and H. Petzeltova, A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Diff. Eq. 199 (2004), 1-21.
11. P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153{191, Springer, Berlin, 2003.
12. P. Fife and X. Wang, A convolution model for interfacial motion: The generation and propagation of internal layers in higher space dimensions, Adv. Diff. Eqs., 3 (1998), 85-110.
13. X. Feng and A. Prohl Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 3 (2003), 35-65.
14. J. Han, The Cauchy problem and steady state solutions for a nonlocal Cahn-Hilliard equation e. j. diff. eqs. 113 (2004) 1-9.
15. T. Kuhn Convergence of a fully discrete approximation for advected mean curvature flows IMA J. Numer. Anal. 18(4), (1998) 595-634.
16. A. de Masi, E. Orlandi, E. Presutti, and L. Triolo, Stability of the interface in a model of phase separation, Proc. Roy. Soc. Edin. A 124 (1994) 1013-1022.
17. A. de Masi, E. Orlandi, E. Presutti, and L. Triolo, Uniqueness and global stability of the instanton in nonlocal evolution equations, Rend. Math. 14 (1994) 693-723.
18. B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. P. D. E. 14 (1989), 245-297.