Slow dynamics for the cahn-hilliard equation in higher space dimensions part i: Spectral estimates

Communications in Partial Differential Equations

Volumn 19, Issue 9-10, 1994, Pages 1397-1447

  Alikakos, Nicholas D ^a,b   Fusco, Giorgio ^c  

^a UNIVERSITY OF TENNESSEE   (United States)

^b UNIVERSITY OF CRETE   (Greece)

^c UNIVERSITY OF ROME TOR VERGATA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84963254790     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605309408821059     Document Type: Article

References (61)

1. [Ag1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations, bounds on eigenfunctions of N-body Schrödinger operators, Mathematical Notes 29, Princeton University Press and University of Tokyo Press, 1982.
2. [Ag2] _, Bounds on exponential decay of eigenfunctions of Schrödinger operator, Lecture Notes in Mathematics, # 1159.
3. [A-B-Fl] N.D. Alikakos, P.W. Bates and G. Fusco, Slow Motion for the Cahn-Hilliard Equation in One Space Dimension, JDE 90 (1991), 81–135.
4. [A-B-F2] _, Slow Motion Manifolds for a class of Singular Perturbation Problems: The Linearized Equations, Differential Equations and Mathematical Physics (C. Bennewitz, eds.), Academic Press, 1991.
5. [A-F1] N.D. Alikakos and G. Fusco, Slow Dynamics for the Cahn-Hithard Equation in Higher Space Dimensions, Part II; preprint.
6. [A-F2] _, The Spectrum of the Linearized Cahn-Hilliard operator for generic interfaces, Indiana Univ. Math. J. 42 (1993), 637–674.
7. [Al-W] F.J.Almgren and L.Wang, Mathematical Existence of crystal growth with Gibbs-Thomson curvature effects, Informal Lecture notes for the workshop on Evolving Phase Boundaries, CMU, March, 1991.
8. [A-McK] N.D. Alikakos and W.R. McKinney, Remarks on the equilibrium theory for the Cahn-Hilliard equation in one space dimension,, Reaction-Diffusion Equations (K.J. Brown and A.A. Lacey, eds.), Oxford Science Publications, 1990, pp. 75–93.
9. [B-F] P.W. Bates and P.C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening, Phys. D 43 (1990), 335–348.
10. [B-X] J. Xun and P.Bates, Metastable Patterns for the Cahn-Hilliard equation Part /; to appear in JDE.
11. [Br] K.A. Brakke, The motion of a surface by its mean curvature, Princeton University Press, Princeton, New Jersey, 1978.
12. [Bro-H] L. Bronsard and D. Hilhorst, On the slow dynamics for the Cahn-Hilliard equation in one space dimension (1992); preprint.
13. [Bro-K1] L. Bronsard and R.V.Kohn, On the slowness of the phase boundary motion in one space dimension, CPAM 43 (1990), 983–998.
14. [Bro-K2] _, Motion by mean curvature as the singular limit of Ginzburgh-Landau dynamics; preprint.
15. [Cal] G. Caginalp, The Dynamics of a conserved Phase Field System: Stefan-like, Hele-Shaw and Cahn-Hilliard Models as Asymptotic limits, IMA J. of Applied Math. 44 (1989), 77–94.
16. [Ca2] _, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A 39 (1989), 5887–5896.
17. [C] J.W. Cahn, On spinodal decomposition, Acta Metallurgica 9 (1961), 795–801.
18. [C-H] J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system I. Interfacial Energy, J T Chemical Physics 28 (1958), 258–256.
19. [Ca] L. Capogna, Proprietá spellrali del'operatore di Schrödinger e stime puntuali sull'autofunzione principale, Thesis, II Universita degli studi di Roma (1990).
20. [C-G-S] J.Carr, M. Gurtin and M. Slemrod, Structured phase transitions on a finite interval, ARMA 86 (1984), 317–357.
21. [C-P1] J. Carr and R.L. Pego, Metastable patterns in solutions of ut = ε2uxx-f(u), CPAM 42 (1989), 523–576.
22. [C-P2] J. Carr and R.L. Pego, Invariant Manifolds for Metastable patterns in ut = ε2uxx-f(u); preprint.
23. [C-Si] R. Carmona and B.Simon, Pointwise Bounds on eigenfunctions and Wave Packets in N-Body Quantum Systems, Comm. Math. Phys. 80 (1981), 59–98.
24. [Ch] X. Chen, Hele-Shaw problem and area preserved curve shortening motion; preprint.
25. [Cou-H] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, John Wiley and Sons, 1989.
26. [C-Gi-Go] Y.G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff.Geom. 33 (1991), 749–786.
27. [deM-S] P. deMottoni and M. Schatzman, Geometrical Evolution of developed interfaces; preprint.
28. [Du-R] J. Duchon and R.Robert, Evolution d'une interface par capillarite et diffusion de volume I. Existence locale en temps, Ann. Inst. Henri Poincare, Analyse non lineaire 1 (1984), 361–378.
29. [E-Fr] C.M. Elliot and D.A. French, Numerical Studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math. 38 (1987), 97–128.
30. [E-S1] L.C. Evans and J.Spruck, Motion of level sets by mean curvature I, J. Diff. Geom 33 (1991), 635–681.
31. [E-S2] _, Motion of level sets by mean curvature II; to appear in Trans. AMS.
32. [E-S3] _, Motion of level sets by mean curvature III; to appear in J.Geom,Anal.
33. [E-S-S] L.C.Evans, H.M.Soner and P.E. Souganidis, Phase Transitions and generalized motion by mean curvature; to appear in CPAM.
34. [Ey] D.Eyre; Ph.D. Thesis, University of Utah, 1992.
35. [F] P.C. Fife, Models for phase separation and their mathematics, Nonlinear Partial Differential Equations and Applications (M. Mimura and Y. Nishiura, eds.), Taniguyia Pubs.
36. [F-G] P.C. Fife and G.S. Gill, The phase-field description of mushy zones, Physica D 35 (1989), 267–275.
37. [F1] G. Folland, Introduction to Partial Differential Equations, Princeton University Press and Tokyo University Press, Princeton, New Jersey, 1976.
38. [Fu-H] G. Fusco and J.K. Hale, Slow motion manifolds, dormant instability and singular perturbation, Dynamics and Differential Equs. I (1989).
39. xx+ f(u) for small ε; Proc. Stuttgart FRG, 1988, Problems Involving Change of Type, K. Kirchgassner (ed.), Lecture Notes in Physics 359.
40. [Ga] M.Gage, On an area preserving evolution equation for plane curves, Contemp. Math 51 (1986), 51–52.
41. [Ga-H] M. Gage and R.S. Hamilton, The heat equation shrinking convex plane curves, J.Diff.Geom. 23 (1986), 69–96.
42. [G1] M.E.Gurtin, On the two-phase Stefan problem with interfacial energy and entropy, ARMA 96 (1988), 199–241.
43. [G2] _, On phase Transitions with bulk, interfacial and boundary energy, ARMA 36 (1986), 234–264.
44. [G3] _, Multiphase thermomechanics with interfacial structured Heat conduction and the capillarity ballance law, ARMA 104 (1988), 195–221.
45. [G4] _, Toward a nonequilibrium thermomechanics of two phase materials, ARMA 100 (1988), 275–312.
46. [G5] _, On the Isothermal Motion of a Phase Interface, Problems Involving Change of Type (V.K. Kirchgassner, ed.), Lecture Notes in Physics 359.
47. [G-M] M. Gurtin and H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quant. Appl. Math. 46 (1988), 301–317.
48. [Gr] C.P. Grant, Slow Motion in one dimensional Cahn-Morral Systems; preprint.
49. [H-Sj] B. Hellfer and S.Sjöstrand, Multiple Wells in the Semi-Classical Limit I, CPDE 9 (1984), 337–408.
50. T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's Motion by Mean Curvature; preprint.
51. [Lu] S.Luckhaus, Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for melting temperature, Euro.Jul.Appl.Math I (1990), 101–111.
52. [Lu-Mod] S. Luckhaus and L. Modica, The Gibbs-Thomson relation with the gradient theory of phase transitions, ARMS 107 (1989), 71–83.
53. [Mod] L. Modica, Gradient theory of phase transitions and singular perturbation, ARMA 98 (1986), 123–142.
54. [McK] W. McKinney; Thesis, Department of Mathematics, University of Tennessee, Knoxville (1989).
55. [Neu] J. Neu; unpublished lecture notes.
56. [Pe] R.L.Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London A 422 (1989), 261–278.
57. [P-F] O. Penrose and P.C.Fife, Thermdynamically consistent models of phase field type for the kinetics of phase transitions, Physica D 43 (1990), 44–62.
58. [S] H.M. Soner, Motion of a set by the curvature of its boundary; preprint.
59. [St] V. Stefanopoulos; Thesis, Department of Mathematics, University of Tennessee at Knoxville, December 1993.
60. [Ste] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.
61. [Ster] P. Sternberg, The effect of a singular perturbation on non-convex variational problems, ARMA 101 (1988), 209–260.