Electronic Research Announcements

Author(s): Bo Su; Martin Burger
Journal: Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 46-52.
MSC (2000): Primary 70H20, 35R35, 35L45
Posted: May 14, 2007
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Abstract: We show the global existence of weak solutions for a free-boundary problem arising in the non-isothermal crystallization of polymers. In particular, the free interface is shown to be of codimension one for every time

in two space dimensions; Hölder continuity of the temperature

is proven.

1.: Avrami, M., Kinetics of phase change I-III, J. Chem. Phys. 7 (1939), 1103-1112; 8 (1940), 212-224; 9 (1941), 177-184.
2.: Barles, G, Soner, H.M., and Souganidis, P.E., Front propagation and phase-field theory, SIAM J. Cont. Optim. 31 (1993), 439-469. MR 1205984 (94c:35005)
3.: Burger, M., Growth and impingement in polymer melts , in: Colli, P., et. al., eds., Free-Boundary Problems, Birkhäuser, Basel, 2003, pp. 65-74. MR 2044564
4.: Burger, M., Growth of multiple crystals in polymer melts, European J. Appl. Math. 15 (2004), 347-363. MR 2092918 (2006c:80003)
5.: Burger, M., and Capasso, V., Mathematical modelling and simulation of non-isothermal crystallization of polymers, Math. Models and Meth. in Appl. Sciences 11 (2001), 1029-1054. MR 1850561 (2002i:82102)
6.: Burger, M., Capasso, V., and Eder, G., Modelling crystallization of polymers in temperature fields, ZAMM 82 (2002), 51-63. MR 1878482 (2003f:82106)
7.: Burger, M., Capasso, V., and Salani, C., Modelling multi-dimensional crystallization of polymers in interaction with heat transfer, Nonlinear Analysis B, Real World Applications 3 (2002), 139-160. MR 1941953 (2004b:82056)
8.: Caffarelli, L.A., and Evans, L.C., Continuity of the temperature in the two-phase Stefan problem, Arch. Rational Mech. Anal. 81 (1983), no. 3, 199-220. MR 683353 (84g:35070)
9.: Capasso, V., and Salani, C., Stochastic birth-and-growth processes modelling crystallization of polymers in a spatially heterogenous temperature field, Nonlinear Analysis, Real World Applications 1 (2000), 485-498. MR 1796584 (2001h:82068)
10.: Chen, G.-Q., and Su, B., Discontinuous solutions in $L\sp\infty$ for Hamilton-Jacobi equations, Chinese Ann. Math. Ser. B 21 (2000), no. 2, 165-186. MR 1763491 (2001c:35052)
11.: Chen, G.-Q., and Su, B., On global discontinuous solutions of Hamilton-Jacobi equations, C. R. Math. Acad. Sci. Paris 334 (2002), no. 2, 113-118. MR 1885091 (2003b:35031)
12.: Chen, G.-Q., and Su, B., Discontinuous solutions for Hamilton-Jacobi equations: uniqueness and regularity, Discrete Contin. Dyn. Syst. 9 (2003), no. 1, 167-192. MR 1951317 (2003k:49062)
13.: Crandall, M.G., and Lions, P.L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1-42. MR 690039 (85g:35029)
14.: DiBenedetto, E., Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl. (4) 130 (1982), 131-176. MR 663969 (83k:35045)
15.: Eder, G., Crystallization kinetic equations incorporating surface and bulk nucleation, ZAMM 76 (1996), S4, 489-492.
16.: Eder, G., Fundamentals of structure formation in crystallizing polymers, in K. Hatada, T. Kitayama, O. Vogl, eds., Macromolecular design of polymeric materials, M. Dekker, New York, 1997, pp. 761-782.
17.: Evans, V.R., The laws of expanding circles and spheres in relations to the lateral growth rate of surface films and the grain-size of metals, Trans. Faraday Soc. 41 (1945), 365-374.
18.: Evans, L.C., and Gariepy, R.F., Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660 (93f:28001)
19.: Friedman, A., Variational principles and free-boundary problems, Wiley and Sons, Inc., New York, 1982. MR 679313 (84e:35153)
20.: Friedman, A., and Velazquez, J.L., A free-boundary problem associated to crystallization of polymers, Indiana Univ. Math. Journal 50 (2001), 1609-1650. MR 1889073 (2003g:35229)
21.: Kolmogorov, A.N., Statistical theory of crystallization of metals, Bull. Acad. Sci. USSR, Math. Ser. 1 (1937), 355-359.
22.: Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural'ceva, N.N., Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. MR 0241822 (39:3159b)
23.: Lions, P.L., Generalized solutions of Hamilton-Jacobi equations, Pitman, Boston, London, Melbourne, 1982. MR 667669 (84a:49038)
24.: Meirmanov, A.M., The Stefan problem, De Gruyter, Berlin, 1992. MR 1154310 (92m:35282)
25.: Micheletti, A., and Burger, M., Stochastic and deterministic simulation of nonisothermal crystallization of polymers, J.Math.Chem. 30 (2001), 169-193. MR 1921620
26.: Micheletti, A., and Capasso, V., The stochastic geometry of polymer crystallization processes, Stoch. Anal. Appl. 15 (1997), 355-373. MR 1454093 (98b:60178)
27.: Monasse, B., and Haudin, J.M., Thermal dependence of nucleation and growth rate in polypropylene by non-isothermal calorimetry, Colloid & Polymer Sci. 264 (1986), 117-122.
28.: Osher, S., and Sethian, J.A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys. 79 (1988), 12-49. MR 965860 (89h:80012)
29.: Ratajski, E., and Janeschitz-Kriegl, H., How to determine high growth speeds in polymer crystallization, Colloid Polym. Sci. 274, (1996), 938-951.
30.: Sacks, P.E., Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), no. 4, 387-409. MR 696738 (84d:35081)
31.: Schulze, G.E.W., and Naujeck, T.R., A growing 2D spherulite and calculus of variations, Colloid & Polymer Science 269 (1991), 689-703.
32.: Su, B., and Burger, M., Weak solutions of a polymer crystal growth model, submitted. See also UCLA CAM Report 06-40, July 2006.
33.: Su, B., and Trivedi, R., in preparation.
34.: Taylor, J.E., Cahn, J.W., and Handwerker, C.A., Geometric models of crystal growth, Acta metall. mater. 40 (1992), 1443-1472.
35.: Trivedi, R., and Teng, J., in preparation.
36.: Visintin, A., Models of phase transitions, Birkhäuser, Boston, 1996. MR 1423808 (98a:80006)
37.: Ziemer, W.P. Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. Amer. Math. Soc. 271 (1982), no. 2, 733-748. MR 654859 (83e:35074)

Retrieve articles in Electronic Research Announcements with MSC (2000): 70H20, 35R35, 35L45

Bo Su
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: bosu@iastate.edu

Martin Burger
Affiliation: Industrial Mathematics Institute, Johannes Kepler University, Altenbergerstr. 69, A 4040 Linz, Austria
Email: martin.burger@jku.at

DOI: 10.1090/S1079-6762-07-00173-4
PII: S 1079-6762(07)00173-4
Keywords: Free boundary, level-set method, heat conduction, growth, crystallization, Hausdorff measure, codimension-one-measure estimate, decomposition
Received by editor(s): September 15, 2006
Posted: May 14, 2007
Communicated by: Luis A. Caffarelli
Copyright of article: Copyright 2007, American Mathematical Society

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