Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

Elisabetta Rocca; Riccarda Rossi

Applications of Mathematics (2008)

  • Volume: 53, Issue: 5, page 485-520
  • ISSN: 0862-7940

Abstract

top
This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature ϑ , an evolution equation for the phase change parameter χ , including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable 𝐮 . The main novelty of the model is that the equations for χ and 𝐮 are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for 𝐮 which needs to be carefully handled. First, we prove a global well-posedness result for the related initial-boundary value problem. Secondly, we address the long-time behavior of the solutions in a simplified situation. We prove that the ω -limit set of the solution trajectories is nonempty, connected and compact in a suitable topology, and that its elements solve the steady state system associated with the evolution problem.

How to cite

top

Rocca, Elisabetta, and Rossi, Riccarda. "Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials." Applications of Mathematics 53.5 (2008): 485-520. <http://eudml.org/doc/37797>.

@article{Rocca2008,
abstract = {This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature $\vartheta $, an evolution equation for the phase change parameter $\chi $, including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable $\{\bf u\}$. The main novelty of the model is that the equations for $\chi $ and $\{\bf u\}$ are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for $\{\bf u\}$ which needs to be carefully handled. First, we prove a global well-posedness result for the related initial-boundary value problem. Secondly, we address the long-time behavior of the solutions in a simplified situation. We prove that the $\omega $-limit set of the solution trajectories is nonempty, connected and compact in a suitable topology, and that its elements solve the steady state system associated with the evolution problem.},
author = {Rocca, Elisabetta, Rossi, Riccarda},
journal = {Applications of Mathematics},
keywords = {nonlinear and degenerating PDE system; global existence; uniqueness; long-time behavior of solutions; $\omega $-limit; phase transitions; thermoviscoelastic materials; nonlinear and degenerating PDE system; global existence; uniqueness; long-time behavior of solutions; -limit},
language = {eng},
number = {5},
pages = {485-520},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials},
url = {http://eudml.org/doc/37797},
volume = {53},
year = {2008},
}

TY - JOUR
AU - Rocca, Elisabetta
AU - Rossi, Riccarda
TI - Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials
JO - Applications of Mathematics
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 5
SP - 485
EP - 520
AB - This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature $\vartheta $, an evolution equation for the phase change parameter $\chi $, including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable ${\bf u}$. The main novelty of the model is that the equations for $\chi $ and ${\bf u}$ are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for ${\bf u}$ which needs to be carefully handled. First, we prove a global well-posedness result for the related initial-boundary value problem. Secondly, we address the long-time behavior of the solutions in a simplified situation. We prove that the $\omega $-limit set of the solution trajectories is nonempty, connected and compact in a suitable topology, and that its elements solve the steady state system associated with the evolution problem.
LA - eng
KW - nonlinear and degenerating PDE system; global existence; uniqueness; long-time behavior of solutions; $\omega $-limit; phase transitions; thermoviscoelastic materials; nonlinear and degenerating PDE system; global existence; uniqueness; long-time behavior of solutions; -limit
UR - http://eudml.org/doc/37797
ER -

References

top
  1. Aizicovici, S., Feireisl, E., 10.1007/PL00001365, J. Evol. Equ. 1 (2001), 69-84. (2001) Zbl0973.35037MR1838321DOI10.1007/PL00001365
  2. Baiocchi, C., Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat. Pura Appl., IV. Ser. 76 (1967), 233-304 Italian. (1967) Zbl0153.17202MR0223697
  3. Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing Leyden (1976). (1976) Zbl0328.47035MR0390843
  4. Bonetti, E., Bonfanti, G., Existence and uniqueness of the solution to a 3D thermoviscoelastic system, Electron. J. Differ. Equ. (2003), Electronic. (2003) Zbl1034.74022MR1971116
  5. Bonetti, E., Bonfanti, G., 10.1155/AAA.2005.105, Abstr. Appl. Anal. 2005 (2005), 105-120. (2005) Zbl1090.74019MR2179438DOI10.1155/AAA.2005.105
  6. Bonetti, E., Bonfanti, G., 10.1016/j.anihpc.2007.05.009, Ann. Inst. H. Poincaré Anal. Non Linéaire (2008), doi: 10.1016/j.anihpc.2007.05.009. (2008) Zbl1152.35505MR2466326DOI10.1016/j.anihpc.2007.05.009
  7. Bonetti, E., Schimperna, G., 10.1007/s00161-003-0152-2, Contin. Mech. Thermodyn. 16 (2004), 319-335. (2004) MR2061321DOI10.1007/s00161-003-0152-2
  8. Bonetti, E., Schimperna, G., Segatti, A., 10.1016/j.jde.2005.04.015, J. Differ. Equations 218 (2005), 91-116. (2005) Zbl1078.74048MR2174968DOI10.1016/j.jde.2005.04.015
  9. Bonfanti, G., Frémond, M., Luterotti, F., Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl. 10 (2000), 1-24. (2000) MR1769184
  10. Brezis, H., Opérateurs maximaux monotones et sémi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, No. 5 North-Holland, Elsevier Amsterdam-London, New York (1973), French. (1973) Zbl0252.47055MR0348562
  11. Colli, P., Laurençot, P., 10.1016/S0167-2789(97)80018-8, Physica D 111 (1998), 311-334. (1998) MR1601442DOI10.1016/S0167-2789(97)80018-8
  12. Feireisl, E., Schimperna, G., 10.1002/mma.659, Math. Methods Appl. Sci. 28 (2005), 2117-2132. (2005) Zbl1079.35021MR2176909DOI10.1002/mma.659
  13. Feireisl, E., Simondon, F., 10.1023/A:1026467729263, J. Dyn. Differ. Equations 12 (2000), 647-673. (2000) Zbl0977.35069MR1800136DOI10.1023/A:1026467729263
  14. Frémond, M., Non-smooth Thermomechanics, Springer Berlin (2002). (2002) Zbl0990.80001MR1885252
  15. Frémond, M., Phase change in mechanics. Lecture notes of XXX Scuola estiva di Fisica Matematica, Ravello, 2005, (to appear). 
  16. Germain, P., Cours de mécanique des milieux continus. Tome I: Théorie générale, Masson Paris (1973), French. (1973) Zbl0254.73001MR0368541
  17. Grasselli, M., Petzeltová, H., Schimperna, G., 10.4171/ZAA/1277, Z. Anal. Anwend. 25 (2006), 51-72. (2006) Zbl1128.35021MR2216881DOI10.4171/ZAA/1277
  18. Horn, W., Sprekels, J., Zheng, S., Global existence of smooth solutions to the Penrose-Fife model for Ising ferromagnets, Adv. Math. Sci. Appl. 6 (1996), 227-241. (1996) Zbl0858.35053MR1385769
  19. Krejčí, P., Rocca, E., Sprekels, J., 10.1112/jlms/jdm032, J. London Math. Soc. 76 (2007), 197-210. (2007) MR2351617DOI10.1112/jlms/jdm032
  20. Krejčí, P., Sprekels, J., Stefanelli, U., 10.1137/S0036141001387604, SIAM J. Math. Anal. 34 (2002), 409-434. (2002) Zbl1034.34053MR1951781DOI10.1137/S0036141001387604
  21. Krejčí, P., Sprekels, J., Stefanelli, U., One-dimensional thermo-viscoplastic processes with hysteresis and phase transitions, Adv. Math. Sci. Appl. 13 (2003), 695-712. (2003) Zbl1049.74036MR2029939
  22. Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars Paris (1969), French. (1969) Zbl0189.40603MR0259693
  23. Luterotti, F., Stefanelli, U., 10.4171/ZAA/1081, Z. Anal. Anwend. 21 (2002), 335-350. (2002) Zbl1003.80003MR1915265DOI10.4171/ZAA/1081
  24. Miranville, A., Zelik, S., 10.1002/mma.464, Math. Methods Appl. Sci. 27 (2004), 545-582. (2004) Zbl1050.35113MR2041814DOI10.1002/mma.464
  25. Nirenberg, L., On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 123 (1959), 115-162. (1959) Zbl0088.07601MR0109940
  26. Rocca, E., Rossi, R., 10.1016/j.jde.2008.02.006, J. Differ. Equations (2008), doi: 10.1016/j.jde.2008.02.006. (2008) Zbl1151.74029MR2460027DOI10.1016/j.jde.2008.02.006
  27. Simon, J., Compact sets in the space L p ( 0 , T ; B ) , Ann. Mat. Pura Appl., IV. Ser. 146 (1987), 65-96. (1987) MR0916688
  28. Simon, L., 10.2307/2006981, Ann. Math. 118 (1983), 525-571. (1983) Zbl0549.35071MR0727703DOI10.2307/2006981
  29. Stefanelli, U., Models of phase change with microscopic movements, PhD. Thesis University of Pavia (2003). (2003) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.