Existence of Dendritic Crystal Growth with Stochastic Perturbations

Journal of Nonlinear Science

Volumn 8, Issue 5, 1998, Pages 491-579

  Yip, N K ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

Dendritic crystal growth; Geometric measure theory; Gibbs Theorem condition; Stochastic analysis; Stochastic heat equation

Indexed keywords

EID: 0346408807     PISSN: 09388974     EISSN: None     Source Type: Journal    
DOI: 10.1007/s003329900059     Document Type: Article

References (46)

1. Allards W., The First Variation of Varifold, Ann. Math., 165 (1976), 199.
2. Almgren F., Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems with Constraints, Mem. Am. Math. Soc., 165 (1976).
3. Almgren R., Variational Algorithms and Pattern Formation in Dendritic Solidification, J. Comput. Phys., 106 (1993), 337-354.
4. Almgren F., Taylor J., Wang L., Curvature Driven Flows: A Variational Approach, SIAM J. Control Optim., 31 (1993), No. 2, 387.
5. Almgren F., Wang L., Mathematical Existence of Crystal Growth with Gibbs-Thomson Curvature Effects, Research Report, Geometry Center, University of Minnesota, 1993 (to appear in J. Geom. Anal.).
6. Bomberi E., Regularity Theory for Almost Minimal Currents, Arch. Rat. Mech. Anal, 78 (1982), 99-130.
7. Chalmers B., Principles of Solidification, John Wiley & Sons, Inc., New York, 1964.
8. Chen X., Reitich F., Local Existence and Uniqueness of Solutions of the Stefan Problem with Surface Tension and Kinetic Undercooling, J. Math. Anal. Appl., 164 (1992), no. 2, 350-362.
9. Da Prato G., Zabczyk J., Stochastic Equations in Infinite Dimensions, Cambridge University Press, New York, 1992.
10. Evans L.C., Gariepy R.F., Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
11. Evans L.C., Partial Differential Equations, Lecture Notes, Center for Pure and Applied Mathematics, Berkeley University, 1994.
12. Federer H., Geometric Measure Theory, Springer-Verlag, New York, 1969.
13. Friedman A., Reitich F., The Stefan Problem with Small Surface Tension, Trans. Am. Math. Soc., 328 (1991), no. 2, 465-515.
14. Funaki T., Singular Limit for Stochastic Reaction-Diffusion Equation and Generation of Random Interfaces, Preprint, Department of Mathematical Sciences, University of Tokyo, 1995.
15. Giusti E., Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1994.
16. Gurtin M.E., Multiphase Thermomechanics with Interfacial Structure. I. Heat Conduction and the Capillary Balance Law, Arch. Rat. Mech. Anal., 104 (1988), 195-221.
17. Gurtin M.E., Thermomechanics of Evolving Phase Boundaries in the Plane, Clarendon Press, Oxford, 1993.
18. Ikeka N., Watanabe S., Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.
19. Karatzas I., Shreve S.E., Brownian Motion and Stochastic Calculus, Second Edition, Springer-Verlag, New York, 1991.
20. Karma A., Fluctuations in Solidification, Phys. Rev. E, 48 (1993), No. 5, 3441-3458.
21. Kessler D.A., Koplik J., Levine H., Pattern Selection in Fingered Growth Phenomena, Adv. Phys., 37 (1988), No. 3, 255-339.
22. Kobayashi R., Numerical Approach to Three-Dimensional Dendritic Solidification, J. Exp. Math., 3 (1994), 59-81.
23. Krylov N.V., Rozovskii B.L., Stochastic Evolution Equations, J. Sov. Math., 16 (1981). 1233-1277.
24. Kuo H.H., Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics, No. 463, Springer-Verlag, New York, 1975.
25. Langer J.S., Instabilities and Pattern Formation in Crystal Growth, Rev. Mod. Phys., 52 (1980), No. 1, 1-28.
26. Langer, J.S., Lectures in the Theory of Pattern Formation, Chance and Matter, ed. J. Soulettie, J. Vannimenus, R. Stora, 629-711, North-Holland, Amsterdam, 1987.
27. Langer J.S., Dendritic Sidebranching in the Three-Dimensional Symmetric Model in the Presence of Noise, Phys. Rev. A, 36 (1987). No. 7, 3350-3358.
28. Luckhaus S., Solutions for Two Phase Stefan Problem with the Gibbs-Thomson Law for the Melting Temperature, Eur. J. Appl. Math., 1 (1990), 101-111.
29. Luckhaus S., Sturzenhecker T., Implicit Time Discretization for the Mean Curvature Flow Equation, Calculus Variations PDE, 3 (1995), 253-271.
30. Metivier M., Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Quaderni, Scuola Normale Superiore, Pisa, 1988.
31. Morgan F., Geometric Measure Theory, A Beginner's Guide, Academic Press, Inc., London, 1988.
32. Mullins W.W., Serkerka R.F., Morphological Stability of a Particle Growing by Diffusion or Heat Flow, J. Appl. Phys., 34 (1963), No. 2, 323-329.
33. Mullins W.W., Serkerka R.F., Stability of a Planar Interface during Solidification of a Dilute Binary Alloy, J. Appl. Phys., 35 (1964), 444-451.
34. Pardoux E., Équations aux Dérivees Partielles Stochastiques Non Linéaires Monotones. Étude de Solutions Fortes de Type Ito, Thesis, A L'Université Paris Sud, Centre D'Osay, 1975.
35. Pieters R., Langer J.S., Noise-Driven Sidebranching in the Boundary Layer Model of Dendritic Solidification, Phys. Rev. Lett., 56 (1986), 1948-1951.
36. Rachev S.T., The Monge-Kantorovich Mass Transference Problem and Its Stochastic Applications, Theory Probab. Appl., XXIX (1984), 647-676.
37. Roosen A.R., Taylor J.E., Modeling Crystal Growth in a Diffusion Field Using Fully Faceted Interfaces, J. Comput. Phys., 114 (1994), 113-128.
38. Soner H.M., Convergence of the Phase Field Equations to the Mullins-Serkerka Problem with Kinetic Undercooling, Arch. Rat. Mech. Anal., 131 (1995), no. 2, 139-197.
39. Strain J., Velocity Effects in Unstable Solidification, SIAM J. Appl. Math., 50 (1990), 1.
40. Stroock D., Varadhan S.R., Multidimensional Diffusion Process, Springer-Verlag, New York, 1979.
41. Taylor J., Cahn J.W., Handwerker C.A., I-Geometric Models of Crystal Growth, Acta Metall. Meter., 40 (1992), 1443-1474.
42. Taylor J., II-Mean Curvature and Weighted Mean Curvature, Acta Metall. Meter., 40 (1992), 1475-1485.
43. Warren J.A., Langer J.S., Prediction of Dendritic Spacings in a Directional-Solidification Experiment, Phys. Rev. E, 47 (1993), No. 4, 2702-2712.
44. Wheeler A.A., Murray B.T., Schaefer R.J., Computation of Dendrites Using a Phase Field Model, Physica D, 66 (1993), 243-262.
45. Woodruff D. P., The Solid-Liquid Interface, Cambridge University Press, London, 1973.
46. Yip N., Stochastic Perturbations in Curvature Driven Flows, Thesis, Princeton University, 1996.