Global weak solutions of non-isothermal front propagation problem
Authors:
Bo Su and Martin Burger
Journal:
Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 46-52
MSC (2000):
Primary 70H20, 35R35, 35L45
DOI:
https://doi.org/10.1090/S1079-6762-07-00173-4
Published electronically:
May 14, 2007
MathSciNet review:
2320681
Full-text PDF Free Access
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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 $t$ in two space dimensions; Hölder continuity of the temperature $u$ is proven.
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MoHa86Monasse, 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.
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RaJa96Ratajski, E., and Janeschitz-Kriegl, H., How to determine high growth speeds in polymer crystallization, Colloid Polym. Sci. 274, (1996), 938–951.
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SchNa91 Schulze, G.E.W., and Naujeck, T.R., A growing 2D spherulite and calculus of variations, Colloid & Polymer Science 269 (1991), 689–703.
SuBu Su, B., and Burger, M., Weak solutions of a polymer crystal growth model, submitted. See also UCLA CAM Report 06-40, July 2006.
SuTr Su, B., and Trivedi, R., in preparation.
TaCaHa92 Taylor, J.E., Cahn, J.W., and Handwerker, C.A., Geometric models of crystal growth, Acta metall. mater. 40 (1992), 1443–1472.
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Av Avrami, M., Kinetics of phase change I-III, J. Chem. Phys. 7 (1939), 1103–1112; 8 (1940), 212–224; 9 (1941), 177–184.
BaSoSo93Barles, G, Soner, H.M., and Souganidis, P.E., Front propagation and phase-field theory, SIAM J. Cont. Optim. 31 (1993), 439–469.
Bu01Burger, M., Growth and impingement in polymer melts , in: Colli, P., et. al., eds., Free-Boundary Problems, Birkhäuser, Basel, 2003, pp. 65–74.
Bu02Burger, M., Growth of multiple crystals in polymer melts, European J. Appl. Math. 15 (2004), 347–363.
BuCa01Burger, 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.
BuCaEd01Burger, M., Capasso, V., and Eder, G., Modelling crystallization of polymers in temperature fields, ZAMM 82 (2002), 51–63.
BuCaSa01Burger, 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.
CaEv83 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.
CaSa00Capasso, 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.
ChSu00Chen, G.-Q., and Su, B., Discontinuous solutions in $L^\infty$ for Hamilton-Jacobi equations, Chinese Ann. Math. Ser. B 21 (2000), no. 2, 165–186.
ChSu02Chen, 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.
ChSu03Chen, G.-Q., and Su, B., Discontinuous solutions for Hamilton-Jacobi equations: uniqueness and regularity, Discrete Contin. Dyn. Syst. 9 (2003), no. 1, 167–192.
CrLi83 Crandall, M.G., and Lions, P.L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1–42.
Di82 DiBenedetto, E., Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl. (4) 130 (1982), 131–176.
Ed96Eder, G., Crystallization kinetic equations incorporating surface and bulk nucleation, ZAMM 76 (1996), S4, 489–492.
Ed97Eder, 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.
Ev45 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.
EvGa92Evans, L.C., and Gariepy, R.F., Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
Fr82 Friedman, A., Variational principles and free-boundary problems, Wiley and Sons, Inc., New York, 1982.
FrVe01Friedman, A., and Velazquez, J.L., A free-boundary problem associated to crystallization of polymers, Indiana Univ. Math. Journal 50 (2001), 1609–1650.
Kol37 Kolmogorov, A.N., Statistical theory of crystallization of metals, Bull. Acad. Sci. USSR, Math. Ser. 1 (1937), 355–359.
LaSoUr67Ladyzhenskaya, 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.
Li82Lions, P.L., Generalized solutions of Hamilton-Jacobi equations, Pitman, Boston, London, Melbourne, 1982.
Me92Meirmanov, A.M., The Stefan problem, De Gruyter, Berlin, 1992.
BuMi00Micheletti, A., and Burger, M., Stochastic and deterministic simulation of nonisothermal crystallization of polymers, J.Math.Chem. 30 (2001), 169–193.
MiCa97Micheletti, A., and Capasso, V., The stochastic geometry of polymer crystallization processes, Stoch. Anal. Appl. 15 (1997), 355–373.
MoHa86Monasse, 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.
OsSe88Osher, S., and Sethian, J.A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations, J. Comp. Phys. 79 (1988), 12–49.
RaJa96Ratajski, E., and Janeschitz-Kriegl, H., How to determine high growth speeds in polymer crystallization, Colloid Polym. Sci. 274, (1996), 938–951.
Sa83 Sacks, P.E., Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), no. 4, 387–409.
SchNa91 Schulze, G.E.W., and Naujeck, T.R., A growing 2D spherulite and calculus of variations, Colloid & Polymer Science 269 (1991), 689–703.
SuBu Su, B., and Burger, M., Weak solutions of a polymer crystal growth model, submitted. See also UCLA CAM Report 06-40, July 2006.
SuTr Su, B., and Trivedi, R., in preparation.
TaCaHa92 Taylor, J.E., Cahn, J.W., and Handwerker, C.A., Geometric models of crystal growth, Acta metall. mater. 40 (1992), 1443–1472.
TrTeng Trivedi, R., and Teng, J., in preparation.
Vi96Visintin, A., Models of phase transitions, Birkhäuser, Boston, 1996.
Zi82Ziemer, W.P. Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. Amer. Math. Soc. 271 (1982), no. 2, 733–748.
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Additional Information
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
Keywords:
Free boundary,
level-set method,
heat conduction,
growth,
crystallization,
Hausdorff measure,
codimension-one-measure estimate,
decomposition
Received by editor(s):
September 15, 2006
Published electronically:
May 14, 2007
Communicated by:
Luis A. Caffarelli
Article copyright:
© Copyright 2007
American Mathematical Society
