Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A nonlocal phase-field system with inertial term

Authors: Maurizio Grasselli, Hana Petzeltová and Giulio Schimperna
Journal: Quart. Appl. Math. 65 (2007), 451-469
MSC (2000): Primary 34D05, 35B40, 35Q99, 80A22
DOI: https://doi.org/10.1090/S0033-569X-07-01070-9
Published electronically: July 19, 2007
MathSciNet review: 2354882
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter $ \chi$ is a nonlocal and nonlinear second-order ODE. The main features of the latter equation are a space convolution term which models long-range interactions of particles and a singular configuration potential that forces $ \chi$ to take values in $ (-1,1)$. We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of $ \omega$-limits. In particular, using a nonsmooth version of the \Lojasiewicz-Simon inequality, we show that the $ \omega$-limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic.

References [Enhancements On Off] (What's this?)

  • 1. H. Amann, ``Linear and Quasilinear Parabolic Problems'', Birkhäuser Verlag, Basel-Boston-Berlin, 1995. MR 1345385 (96g:34088)
  • 2. V. Barbu, ``Nonlinear Semigroups and Differential Equations in Banach Spaces'', Noordhoff, Leyden, 1976. MR 0390843 (52:11666)
  • 3. P.W. Bates, F. Chen, Traveling wave solutions for a nonlocal phase-field system, Interfaces Free Bound., 4 (2002), 227-238. MR 1914622 (2003f:35275)
  • 4. P.W. Bates, F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation, J. Math. Anal. Appl., 273 (2002), 45-57. MR 1933014 (2003h:35104)
  • 5. P.W. Bates, J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differential Equations, 212 (2005), 235-277. MR 2129092 (2005m:35141)
  • 6. M. Brokate, J. Sprekels, ``Hysteresis and Phase Transitions'', Springer, New York, 1996. MR 1411908 (97g:35127)
  • 7. G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. MR 816623 (87c:80011)
  • 8. J.C. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
  • 9. R. Chill, On the \Lojasiewicz-Simon gradient inequality, J. Funct. Anal., 201 (2003), 572-601. MR 1986700 (2005c:26019)
  • 10. R. Chill, M.A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2003), 1017-1039. MR 1978032 (2004d:34103)
  • 11. E. Feireisl, F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Differential Equations, 12 (2000), 647-673. MR 1800136 (2002g:35116)
  • 12. E. Feireisl, F. Issard-Roch, H. Petzeltová, A non-smooth version of the \Lojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21. MR 2041509 (2005c:35284)
  • 13. H. Gajewski, On a nonlocal model of non-isothermal phase separation, Adv. Math. Sci. Appl., 12 (2002), 569-586. MR 1943981 (2003k:35103)
  • 14. H. Gajewski, K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31. MR 2009615 (2004i:35163)
  • 15. P. Galenko, D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125(13).
  • 16. G. Giacomin, J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, J. Statist. Phys., 87 (1997), 37-61. MR 1453735 (98m:82053)
  • 17. G. Giacomin, J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion, SIAM J. Appl. Math., 58 (1998), 1707-1729 (electronic). MR 1638739 (99m:35249)
  • 18. C. Giorgi, M. Grasselli and V. Pata, Uniform attractors for a phase field model with memory and quadratic nonlinearity, Indiana Univ. Math. J., 48 (1999), 1395-1445. MR 1757078 (2001h:37160)
  • 19. M. Grasselli, A. Miranville, V. Pata, S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., to appear.
  • 20. M. Grasselli, V. Pata, Existence of a universal attractor for a parabolic-hyperbolic phase-field system, Adv. Math. Sci. Appl., 13 (2003), 443-459. MR 2029927 (2004k:37172)
  • 21. M. Grasselli, V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal., 3 (2004), 849-881. MR 2106302 (2005h:35150)
  • 22. M. Grasselli, H. Petzeltová, G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwendungen, 25 (2006), 51-72. MR 2216881 (2007b:35159)
  • 23. M. Grasselli, H. Petzeltová, G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system, Commun. Pure Appl. Anal., 5 (2006), 827-838. MR 2246010
  • 24. S.-Z. Huang, P. Takác, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal., 46 (2001) 675-698. MR 1857152 (2002f:35125)
  • 25. M.A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. MR 1609269 (99c:35101)
  • 26. P. Krejcí, J. Sprekels, Nonlocal phase-field models for non-isothermal phase transitions and hysteresis, Adv. Math. Sci. Appl., 14 (2004), 593-612. MR 2111831 (2006d:74063)
  • 27. P. Krejcí, J. Sprekels, Long time behaviour of a singular phase transition model, Discrete Contin. Dyn. Syst., 15 (2006), 1119-1135. MR 2224500
  • 28. P. Krejcí, E. Rocca, J. Sprekels, Nonlocal temperature-dependent phase-field models for non-isothermal phase transitions, WIAS preprint n. 1006, Berlin (2005).
  • 29. S. \Lojasiewicz, Une propriété topologiqque des sous-ensembles analytiques réels, in Colloques internationaux du C.N.R.S. 117: Les équations aux dérivées partielles (Paris, 1962), 87-89. Editions du C.N.R.S., Paris, 1963. MR 0160856 (28:4066)
  • 30. S. \Lojasiewicz, ``Ensembles Semi-analytiques'', notes, I.H.E.S., Bures-sur-Yvette, 1965.
  • 31. L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math. (2), 118 (1983), 525-571. MR 727703 (85b:58121)
  • 32. J. Sprekels, S. Zheng, Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions, J. Math. Anal. Appl., 279 (2003), 97-110. MR 1970493 (2004c:45015)
  • 33. X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183 (2002), 434-461. MR 1919786 (2003f:35157)
  • 34. H. Wu, M. Grasselli, S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions, Math. Models Methods Appl. Sci., 17 (2007), 125-153. MR 2290411
  • 35. S. Zheng, ``Nonlinear Evolution Equations'', Chapman & Hall/CRC, Boca Raton, Florida, 2004. MR 2088362 (2006a:35001)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 34D05, 35B40, 35Q99, 80A22

Retrieve articles in all journals with MSC (2000): 34D05, 35B40, 35Q99, 80A22

Additional Information

Maurizio Grasselli
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, I-20133 Milano, Italy
Email: maurizio.grasselli@polimi.it

Hana Petzeltová
Affiliation: Mathematical Institute AS CR, Žitná, 25, CZ-115 67, Praha 1, Czech Republic
Email: petzelt@math.cas.cz

Giulio Schimperna
Affiliation: Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1, I-27100 Pavia, Italy
Email: giusch04@unipv.it

DOI: https://doi.org/10.1090/S0033-569X-07-01070-9
Received by editor(s): May 8, 2006
Published electronically: July 19, 2007
Additional Notes: This work was partially supported by the Italian PRIN Research Project Problemi a frontiera libera, transizioni di fase e modelli di isteresi
The work of the second author was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan no. AV0Z10190503 and by Grant IAA1001190606 of GA AV ČR
The work of the last author was partially supported by the HYKE Research Training Network
Article copyright: © Copyright 2007 Brown University

American Mathematical Society