Skip to main content
SpringerLink
Log in

On a Class of Conserved Phase Field Systems with a Maximal Monotone Perturbation

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

We prove existence and regularity for the solutions to a Cahn–Hilliard system describing the phenomenon of phase separation for a material contained in a bounded and regular domain. Since the first equation of the system is perturbed by the presence of an additional maximal monotone operator, we show our results using suitable regularization of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments. Next, under further assumptions, we deduce a continuous dependence estimate whence the uniqueness property is also achieved. Then, we consider the related sliding mode control (SMC) problem and show that the chosen SMC law forces a suitable linear combination of the temperature and the phase to reach a given (space-dependent) value within finite time.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, New York (2010)

    Book  Google Scholar 

  2. Barbu, V., Colli, P., Gilardi, G., Marinoschi, G., Rocca, E.: Sliding mode control for a nonlinear phase-field system. SIAM J. Control Optim, to appear 2017 (see also preprint arXiv:1506.01665 [math.AP]) pp. 1–28 (2015)

  3. Brezis, H.: “Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert”, North-Holland Math. Stud. 5, North-Holland, Amsterdam (1973)

  4. Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Springer, New York (1996)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system I. Interfacial free energy J. Chem. Phys. 2, 258–267 (1958)

    Google Scholar 

  7. Cherfils, L., Miranville, A., Zelik, S.: The Cahn-Hilliard equation with logarithmic potentials. Milan J. Math. 79, 561–596 (2011)

    Article  MathSciNet  Google Scholar 

  8. Colli, P., Farshbaf-Shaker, M.H., Gilardi, G., Sprekels, J.: Optimal boundary control of a viscous Cahn–Hilliard system with dynamic boundary condition and double obstacle potentials. SIAM J. Control Optim. 53, 2696–2721 (2015)

    Article  MathSciNet  Google Scholar 

  9. Colli, P., Farshbaf-Shaker, M.H., Gilardi, G., Sprekels, J.: Second-order analysis of a boundary control problem for the viscous CahnHilliard equation with dynamic boundary condition. Ann. Acad. Rom. Sci. Ser. Math. Appl. 7, 41–66 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Colli, P., Fukao, T.: Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary. J. Math. Anal. Appl. 429, 1190–1213 (2015)

    Article  MathSciNet  Google Scholar 

  11. Colli, P., Fukao, T.: Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials. Nonlinear Anal. 127, 413–433 (2015)

    Article  MathSciNet  Google Scholar 

  12. Colli, P., Fukao, T.: Nonlinear diffusion equations as asymptotic limits of CahnHilliard systems. J. Differ. Equ. 260, 6930–6959 (2016)

    Article  Google Scholar 

  13. Colli, P., Gilardi, G., Marinoschi, G.: A boundary control problem for a possibly singular phase field system with dynamic boundary conditions. J. Math. Anal. Appl. 434, 432–463 (2016)

    Article  MathSciNet  Google Scholar 

  14. Colli, P., Gilardi, G., Marinoschi, G., Rocca, E.: Optimal control for a phase field system with a possibly singular potential. Math. Control Relat. Fields 6, 95–112 (2016)

    Article  MathSciNet  Google Scholar 

  15. Colli, P., Gilardi, G., Sprekels, J.: On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential. J. Math. Anal. Appl. 419, 972–994 (2014)

    Article  MathSciNet  Google Scholar 

  16. Colli, P., Gilardi, G., Sprekels, J.: A boundary control problem for the pure CahnHilliard equation with dynamic boundary conditions. Adv. Nonlinear Anal. 4, 311–325 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Colli, P., Gilardi, G., Sprekels, J.: A boundary control problem for the viscous CahnHilliard equation with dynamic boundary conditions. Appl. Math. Optim. 73, 195–225 (2016)

    Article  MathSciNet  Google Scholar 

  18. Colli, P., Gilardi, G., Sprekels, J.: Constrained evolution for a quasilinear parabolic equation. J. Optim. Theory Appl. 170, 713–734 (2016)

    Article  MathSciNet  Google Scholar 

  19. Colli, P., Laurençot, P., Sprekels, J.: Global solution to the Penrose–Fife phase field model with special heat flux laws. Variations of Domain and Free-Boundary Problems in Solid Mechanics, Solid Mech. Appl, vol. 66, pp. 181–188. Kluwer, Dordrecht (1999)

  20. Colli, P., Marinoschi, G., Rocca, E.: Sharp interface control in a Penrose-Fife model. ESAIM Control Optim. Calc. Var. 22, 473–499 (2016)

    Article  MathSciNet  Google Scholar 

  21. Colturato, M.: Solvability of a class of phase field systems related to a sliding mode control problem. Appl. Math. 6, 623–650 (2016)

    Article  MathSciNet  Google Scholar 

  22. Elliott, C.M., Zheng, S.: On the Cahn-Hilliard equation. Arch. Rational Mech. Anal. 96, 339–357 (1986)

    Article  MathSciNet  Google Scholar 

  23. Elliott, C.M., Zheng, S.: Global existence and stability of solutions to the phase-field equations. Internat. Ser. Numer. Math. 95, pp. 46–58, in “Free boundary problems”, Birkhäuser Verlag, Basel (1990)

  24. Gilardi, G., Miranville, A., Schimperna, G.: On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Commun. Pure Appl. Anal. 8, 881–912 (2009)

    Article  MathSciNet  Google Scholar 

  25. Grasselli, M., Petzeltová, H., Schimperna, G.: Long time behavior of solutions to the Caginalp system with singular potential. Z. Anal. Anwend. 25, 51–72 (2006)

    Article  MathSciNet  Google Scholar 

  26. Heida, M.: Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation. Appl. Math. 60, 51–90 (2015)

    Article  MathSciNet  Google Scholar 

  27. Hintermüller, M., Wegner, D.: Distributed optimal control of the Cahn-Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 50, 388–418 (2012)

    Article  MathSciNet  Google Scholar 

  28. Hintermüller, M., Wegner, D.: Optimal control of a semi-discrete Cahn-Hilliard-Navier stokes system. SIAM J. Control Optim. 52, 747–772 (2014)

    Article  MathSciNet  Google Scholar 

  29. Hoffmann, K.H., Kenmochi, N., Kubo, M., Yamazaki, N.: Optimal control problems for models of phase-field type with hysteresis of play operator. Adv. Math. Sci. Appl. 17, 305–336 (2007)

    MathSciNet  MATH  Google Scholar 

  30. Itkis, U.: Control Systems of Variable Structure. Wiley, Hoboken (1976)

    Google Scholar 

  31. Kenmochi, N., Niezgódka, M.: Evolution systems of nonlinear variational inequalities arising from phase change problems. Nonlinear Anal. 22, 1163–1180 (1994)

    Article  MathSciNet  Google Scholar 

  32. Kenmochi, N., Niezgódka, M.: Viscosity approach to modelling non-isothermal diffusive phase separation. Japan J. Ind. Appl. Math. 13, 135–169 (1996)

    Article  MathSciNet  Google Scholar 

  33. Kubo, M.: The Cahn-Hilliard equation with time-dependent constraint. Nonlinear Anal. 75, 5672–5685 (2012)

    Article  MathSciNet  Google Scholar 

  34. Laurençot, Ph: Long-time behaviour for a model of phase-field type. Proc. R. Soc. Edinb. Sect. A 126, 167–185 (1996)

    Article  MathSciNet  Google Scholar 

  35. Orlov, Y.V.: Discontinuous unit feedback control of uncertain infinitedimensional systems. IEEE Trans. Autom. Control 45, 834–843 (2000)

    Article  Google Scholar 

  36. Pruss, J., Racke, R., Zheng, S.: Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions. Ann. Mat. Pura Appl. 185(4), 627–648 (2006)

    Article  MathSciNet  Google Scholar 

  37. Racke, R., Zheng, S.: The Cahn-Hilliard equation with dynamic boundary conditions. Adv. Differ. Equ. 8, 83–110 (2003)

    MathSciNet  MATH  Google Scholar 

  38. Showalter, R.E.: Monotone operators in Banach Space and Nonlinear Partial Differential Equations, AMS, Mathematica Surveys and Monographs, vol. 49. AMS, providence (1991)

  39. Simon, J.: Compact sets in the spaces \(L^p(0,T;B)\). Ann. Mat. Pura. Appl. 146(4), 65–96 (1987)

    MathSciNet  MATH  Google Scholar 

  40. Sprekels, J., Zheng, S.M.: Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions. J. Math. Anal. Appl. 176, 200–223 (1993)

    Article  MathSciNet  Google Scholar 

  41. Wang, Q.F.: Optimal distributed control of nonlinear Cahn–Hilliard systems with computational realization. J. Math. Sci. (N. Y.) 177, 440–458 (2011)

    Article  MathSciNet  Google Scholar 

  42. Wu, H., Zheng, S.: Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions. J. Differ. Equ. 204, 511–531 (2004)

    Article  MathSciNet  Google Scholar 

  43. Zhao, X.P., Liu, C.C.: Optimal control of the convective Cahn-Hilliard equation. Appl. Anal. 92, 1028–1045 (2013)

    Article  MathSciNet  Google Scholar 

  44. Zhao, X.P., Liu, C.C.: Optimal control for the convective Cahn-Hilliard equation in 2D case. Appl. Math. Optim. 70, 61–82 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is very grateful to Professor Pierluigi Colli for his advice, for his kind helpfulness and for several precious discussions. Moreover, MC expresses his gratitude to the referee for the careful reading of the manuscript and for a number of useful suggestions. Some partial support from the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM is acknowledged.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di, Pavia Via Ferrata 1, 27100, Pavia, Italy

    Michele Colturato

Authors
  1. Michele Colturato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michele Colturato.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Colturato, M. On a Class of Conserved Phase Field Systems with a Maximal Monotone Perturbation. Appl Math Optim 78, 545–585 (2018). https://doi.org/10.1007/s00245-017-9415-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-017-9415-3

Keywords

Mathematics Subject Classification

Search

Navigation