On a mathematical model for the crystallization of polymers

Maria Pia Gualdani

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 1, page 161-179
  • ISSN: 0392-4041

Abstract

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We consider a mathematical model proposed in [1] for the cristallization of polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. The model includes a constraint W e ⁢ q on the crystal volume fraction. Essentially, the model is a system of both second order and first order evolutionary partial differential equations with nonlinear terms which are Lipschitz continuous, as in [1], or Hölder continuous, as in [3]. The main novelty here is the fact that W e ⁢ q is a function depending on the temperature T (which is actually the case). We analyse the model in two different conditions: for constitutive equations of non-Lipschitz type, we use monotonicity and L 1 -technique to prove existence and continuous dependence on the data of a weak solution. For more regular constitutive equations, using a fixed point tecnique, we prove a global existence and uniqueness result for a classical solution.

How to cite

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Gualdani, Maria Pia. "On a mathematical model for the crystallization of polymers." Bollettino dell'Unione Matematica Italiana 6-B.1 (2003): 161-179. <http://eudml.org/doc/195004>.

@article{Gualdani2003,
abstract = {We consider a mathematical model proposed in [1] for the cristallization of polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. The model includes a constraint $W_\{eq\}$ on the crystal volume fraction. Essentially, the model is a system of both second order and first order evolutionary partial differential equations with nonlinear terms which are Lipschitz continuous, as in [1], or Hölder continuous, as in [3]. The main novelty here is the fact that $W_\{eq\}$ is a function depending on the temperature T (which is actually the case). We analyse the model in two different conditions: for constitutive equations of non-Lipschitz type, we use monotonicity and $L^\{1\}$-technique to prove existence and continuous dependence on the data of a weak solution. For more regular constitutive equations, using a fixed point tecnique, we prove a global existence and uniqueness result for a classical solution.},
author = {Gualdani, Maria Pia},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {161-179},
publisher = {Unione Matematica Italiana},
title = {On a mathematical model for the crystallization of polymers},
url = {http://eudml.org/doc/195004},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Gualdani, Maria Pia
TI - On a mathematical model for the crystallization of polymers
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/2//
PB - Unione Matematica Italiana
VL - 6-B
IS - 1
SP - 161
EP - 179
AB - We consider a mathematical model proposed in [1] for the cristallization of polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. The model includes a constraint $W_{eq}$ on the crystal volume fraction. Essentially, the model is a system of both second order and first order evolutionary partial differential equations with nonlinear terms which are Lipschitz continuous, as in [1], or Hölder continuous, as in [3]. The main novelty here is the fact that $W_{eq}$ is a function depending on the temperature T (which is actually the case). We analyse the model in two different conditions: for constitutive equations of non-Lipschitz type, we use monotonicity and $L^{1}$-technique to prove existence and continuous dependence on the data of a weak solution. For more regular constitutive equations, using a fixed point tecnique, we prove a global existence and uniqueness result for a classical solution.
LA - eng
UR - http://eudml.org/doc/195004
ER -

References

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  9. FASANO, A., Modelling the solidification of polymers; an example of an ECMI cooperation, ICIAM 91, R.E. O'Malley ed. (1991), 99-118. 
  10. FASANO, A.- MANCINI, A.- MAZZULLO, S., Isobaric crystallization of polypropylene, «Complex Flow in Industrial Processes», A. Fasano ed., Birkhäuser, 5 (2000), 149-190. Zbl0943.76087MR1738012
  11. FASANO, A.- PRIMICERIO, M., On a mathematical models for nucleation and crystal growth processes, Boundary Value Problems for P.D.E.'s and Applications, J. L. Lions, C. Baiocchi eds., Masson (1993), 351-358. Zbl0810.35160MR1260462
  12. LADYZENSKAJA, O. A.- SOLONNIKOV, V. A.- URAL'CEVA, N. N., Linear and quasi-linear Equation of Parabolic Type, translation of Mathematical Monographs, vol. 23, Providence R.I., American Mathematical Society (1968). Zbl0174.15403MR241821
  13. LIONS, J. L.- MAGENES, E., Non-Homogeneous Boundary Value Problems and Application, I Springer-Verlag, Berlin, 1972. Zbl0223.35039
  14. MANCINI, A., Non-isothermal crystallization of polypropylene, in preparation. 
  15. FASANO, A.- MANCINI, A., Existence and uniqueness of a classical solution for a mathematical model describing the isobaric crystallization of a polymer, Interfaces & Free Boundaries, 2 (2000), 1-19. Zbl0949.35146
  16. SOLONNIKOV, V. A., Solvability of the classical initial boundary value problem for the heat conduction equation in a dihedral angle, J. Soviet Math., 32 (1986), 526-546. Zbl0582.35055

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