Skip to main content
SpringerLink
Log in

Global Attractors for the Quasistationary Phase Field Model: a Gradient Flow Approach

  • Conference paper
  • First Online:
Free Boundary Problems

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 154))

  • 1536 Accesses

Abstract

In this note we summarize some results of a forthcoming paper (see [15]), where we examine, in particular, the long time behavior of the so-called quasistationary phase field model by using a gradient flow approach. Our strategy in fact, is inspired by recent existence results which show that gradient flows of suitable non-convex functionals yield solutions to the related quasistationary phase field systems. Thus, we firstly present the long-time behavior of solutions to an abstract non-convex gradient flow equation, by carefully exploiting the notion of generalized semiflows by J.M. Ball and we provide some sufficient conditions for the existence of the global attractor for the solution semiflow. Then, the existence of the global attractor for a proper subset of all the solutions to the quasistationary phase field model is obtained as a byproduct of our abstract results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997), 475–502.

    Article  MathSciNet  MATH  Google Scholar 

  2. _____, Global attractors for damped semilinear wave equations, Discrete Continuous. Dynam. Systems, 10(1) (2004), 31–52.

    MathSciNet  MATH  Google Scholar 

  3. J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for the phase separations with nonsmooth free energy. I, European J. Appl. Math, 2(3) (1991), 233–280.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 50 #1060

    MATH  Google Scholar 

  5. M. Brokate and J. Sprekels, Hysteresis and phase transitions, Appl. Math. Sci., 121, Springer, New York, 1996.

    Google Scholar 

  6. G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92, (1986), 205–245.

    Article  MathSciNet  MATH  Google Scholar 

  7. V.V. Chepyshoz and M.I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002.

    Google Scholar 

  8. P. Colli and P.I. Plotnikov, Global solution to a quasistationary Penrose-Fife model, Indiana Univ. Math. J., 54(2) (2005), 349–382.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Haraux, Systèmes dynamiques dissipatifs et applications, Recherches en Mathématiques Appliquées, Masson, Paris 1991.

    MATH  Google Scholar 

  10. V.S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal. 6(4) (1998), 83–111.

    Article  MathSciNet  MATH  Google Scholar 

  11. P.I. Plotnikov and V.N. Starovoitov, The Stefan problem with surface tension as the limit of a phase field model, Differential Equations 29 (1993), 395–404.

    MathSciNet  MATH  Google Scholar 

  12. E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems, Phys. D 192(3–4) (2004), 279–307.

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Rossi and G. Savaré, Existence and approximation results for gradient flows, “Rend. Mat. Acc. Lincei”, 15 (2004), 183–196.

    MathSciNet  MATH  Google Scholar 

  14. R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, preprint IMATI-CNR n. 7-PV (2004) 1–45. To appear on ESAIM Control Optim. Calc. Var.

    Google Scholar 

  15. R. Rossi, A. Segatti and U. Stefanelli, Attractors for gradient flows of non convex functionals and applications, (2005). Submitted.

    Google Scholar 

  16. R. Schätzle, The quasistationary phase field equations with Neumann boundary conditions, J. Differential Equations, 162(2) (2000), 473–503. MR 2001b:35143

    Article  MathSciNet  MATH  Google Scholar 

  17. G.R. Sell, Differential equations without uniqueness and classical topological dynamics, J. Differential Equations 14 (1973), 42–56.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Segatti, Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Contin. Dyn. Syst. 14(4) (2006), 801–820.

    Article  MathSciNet  MATH  Google Scholar 

  19. K. Shirakawa, A. Ito, N. Yamazaki, and N. Kenmochi, Asymptotic stability for evolution equations governed by subdifferentials. Recent developments in domain decomposition methods and flow problems (Kyoto, 1996; Anacapri, 1996), GAKUTO Internat. Ser. Math. Sci. Appl., 11, 287–310, Gakkōtosho, Tokyo, 1998.

    MathSciNet  MATH  Google Scholar 

  20. R. Temam, Infinite dimensional mechanical systems in mechanics and physics. Applied Mathematical Sciences 68, Springer-Verlag, New York, 1988. 1988.

    Google Scholar 

  21. A. Visintin, Models of phase transitions, Progress in Nonlinear Differential Equations and Their Applications, vol. 28, Birkhäuser, Boston, 1996.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics “F. Casorati”, Università di Pavia, Via Ferrata 1, I-27100, Pavia, Italy

    Antonio Segatti

Authors
  1. Antonio Segatti
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Apartado 3008, 3001-454, Coimbra, Portugal

    Isabel Narra Figueiredo

  2. Universidade de Lisboa / CMAF, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal

    José Francisco Rodrigues

  3. Departamento de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Lisa Santos

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Segatti, A. (2006). Global Attractors for the Quasistationary Phase Field Model: a Gradient Flow Approach. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_37

Download citation

Publish with us

Policies and ethics

Search

Navigation