Hostname: page-component-669899f699-2mbcq Total loading time: 0 Render date: 2025-04-28T01:44:35.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Allen–Cahn equation with strong irreversibility

Published online by Cambridge University Press:  16 July 2018

GORO AKAGI  and
MESSOUD EFENDIEV
GORO AKAGI
Affiliation:
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan Helmholtz Zentrum München, Institut für Computational Biology, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, D-85748 Garching, Germany email: goro.akagi@tohoku.ac.jp
MESSOUD EFENDIEV
Affiliation:
Helmholtz Zentrum München, Institut für Computational Biology, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany email: messoud.efendiyev@helmholtz-muenchen.de
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with a fully non-linear variant of the Allen–Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. The main purposes of this paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviours of solutions. More precisely, by deriving partial energy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solution u(x,t) converges to a solution of an elliptic obstacle problem as t → +∞.

Keywords

Strongly irreversible evolution equationAllen–Cahn equationobstacle parabolic problemglobal attractorω-limit setpartial energy-dissipation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 4 , August 2019 , pp. 707 - 755
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

† G. Akagi is supported in part by JSPS KAKENHI Grant Numbers JP16H03946, JP16K05199, JP17H01095, in part by the Alexander von Humboldt Foundation and in part by the Carl Friedrich von Siemens Foundation.

References

[1] Akagi, G. (2014) Local solvability of a fully nonlinear parabolic equation. Kodai Math. J. 37, 702–727.Google Scholar
[2] Akagi, G. & Efendiev, M. Lyapunov stability of non-isolated equilibria for strongly irreversible Allen–Cahn equations.Google Scholar
[3] Akagi, G. & Kimura, M. (in press) Unidirectional evolution equations of diffusion type. J. Differ. Eq.Google Scholar
[4] Ambrosio, L. & Tortorelli, V. M. (1990) Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Commun. Pure Appl. Math. 43, 999–1036.Google Scholar
[5] Ambrosio, L. & Tortorelli, V. M. (1992) On the approximation of free discontinuity problems. Boll. dell'Unione Math. Italiana 6-B, 105–123.Google Scholar
[6] Arai, T. (1979) On the existence of the solution for ∂ ϕ(u'(t)) + ∂ ψ(u(t)) ∋ f(t). J. Faculty Sci. Univ. Tokyo Sect. IA Math. 26, 75–96.Google Scholar
[7] Aso, M., Frémond, M. & Kenmochi, N. (2005) Phase change problems with temperature dependent constraints for the volume fraction velocities. Nonlinear Anal. 60, 1003–1023.Google Scholar
[8] Aso, M. & Kenmochi, N. (2005) Quasivariational evolution inequalities for a class of reaction–diffusion systems. Nonlinear Anal. 63, e1207–e1217.Google Scholar
[9] Attouch, H. (1984) Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA.Google Scholar
[10] Babadjian, J.-F. & Millot, V. (2014) Unilateral gradient flow of the Ambrosio-Tortorelli functional by minimizing movements. Ann. de l'Institut Henri Poincaré (C) Non Linear Anal. 31, 779–822.Google Scholar
[11] Babin, A. V. & Vishik, M. I. (1992) Attractors of Evolution Equations, Studies in Mathematics and Its Applications, Vol. 25, North-Holland: North-Holland Publishing Co., Amsterdam.Google Scholar
[12] Barbu, V. (1976) Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff: Noordhoff International Publishing, Leiden.Google Scholar
[13] Barbu, V. (1975) Existence theorems for a class of two point boundary problems. J. Differ. Eq. 17, 236–257.Google Scholar
[14] Barenblatt, G. I. & Prostokishin, V. M. (1993) A mathematical model of damage accumulation taking into account microstructural effects. Eur. J. Appl. Math. 4, 225–240.Google Scholar
[15] Bertsch, M. & Bisegna, P. (1997) Blow-up of solutions of a nonlinear parabolic equation in damage mechanics. Eur. J. Appl. Math. 8, 89–123.Google Scholar
[16] Bonetti, E. & Schimperna, G. (2004) Local existence for Frémond's model of damage in elastic materials. Contin. Mech. Thermodyn. 16, 319–335.Google Scholar
[17] Bonfanti, G., Frémond, M. & Luterotti, F. (2000) Global solution to a nonlinear system for irreversible phase changes. Adv. Math. Sci. Appl. 10, 1–24.Google Scholar
[18] Bonfanti, G., Frémond, M. & Luterotti, F. (2001) Local solutions to the full model of phase transitions with dissipation. Adv. Math. Sci. Appl. 11, 791–810.Google Scholar
[19] Brézis, H., Crandall, M. G. & Pazy, A. (1970) Perturbations of nonlinear maximal monotone sets in Banach space. Commun. Pure. Appl. Math. 23, 123–144.Google Scholar
[20] Brézis, H. (1973) Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Math Studies, Vol. 5 North-Holland, Amsterdam/New York.Google Scholar
[21] Brézis, H. (1971) Monotonicity methods in Hilbert spaces and some applications to non-linear partial differential equations. In Zarantonello, E. (editor), Contributions to Nonlinear Functional Analysis, Academic Press, New York, pp. 101–156.Google Scholar
[22] Caffarelli, L. A. (1998) The obstacle problem revisited. J. Fourier Anal. Appl. 4, 383–402.Google Scholar
[23] Caffarelli, L. & Figalli, A. (2013) Regularity of solutions to the parabolic fractional obstacle problem. J. Reine Angewandte Math. 680, 191–233.Google Scholar
[24] Colli, P. (1992) On some doubly nonlinear evolution equations in Banach spaces. Japan J. Ind. Appl. Math. 9, 181–203.Google Scholar
[25] Colli, P. & Visintin, A. (1990) On a class of doubly nonlinear evolution equations. Commun. Partial Differ. Eq. 15, 737–756.Google Scholar
[26] Dal Maso, G. & Toader, R. (2002) A model for the quasi-static growth of brittle fractures: Existence and approximation results. Archive Ration. Mech. Anal. 162, 101–135.Google Scholar
[27] Efendiev, M. (2010) Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics, GAKUTO International Series, Mathematical Sciences and Applications, Vol. 33, Gakkōtosho, Tokyo.Google Scholar
[28] Efendiev, M. (2009) Fredholm Structures, Topological Invariants and Applications, AIMS Series on Differential Equations & Dynamical Systems, Vol. 3, American Institute of Mathematical Sciences (AIMS), Springfield, MO.Google Scholar
[29] Efendiev, M. & Mielke, A. (2006) On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13, 151–167.Google Scholar
[30] Engelking, R. (1977) General Topology (translated from Polish by the author), Monografie Matematyczne, Tom 60 [Mathematical Monographs, Vol. 60], PWN – Polish Scientific Publishers, Warsaw.Google Scholar
[31] Francfort, G. A. (2006) Quasistatic brittle fracture seen as an energy minimizing movement. GAMM-Mitteilungen 29, 172–191.Google Scholar
[32] Francfort, G. A. & Larsen, C. J. (2003) Existence and convergence for quasi-static evolution in brittle fracture. Commun. Pure Appl. Math. 56, 1465–1500.Google Scholar
[33] Francfort, G. A. & Marigo, J.-J. (1998) Revisiting brittle fractures as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342.Google Scholar
[34] Giacomini, A. (2005) Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures. Calculus Var. Partial Differ. Eq. 22, 129–172.Google Scholar
[35] Gianazza, U. & Savaré, G. (1994) Some results on minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Math. (5) 18, 57–80.Google Scholar
[36] Gianazza, U., Gobbino, M. & Savaré, G. (1994) Evolution problems and minimizing movements. Atti Accad. Naz. Lincei Cl. Sci. Fis. Math. Natur. Rend. Lincei (9) Math. Appl 5, 289–296.Google Scholar
[37] Kachanov, L. M. (1986) Introduction to Continuum Damage Mechanics, Mechanics of Elastic Stability, Vol. 10, Springer, Netherlands.Google Scholar
[38] Kimura, M. & Takaishi, T. (2011) Phase field models for crack propagation. Theor. Appl. Mech. Japan 59, 85–90.Google Scholar
[39] Kinderlehrer, D. & Stampacchia, G. (1980) An Introduction to Variational Inequalities and their Applications, Pure and Applied Mathematics, Vol. 88, Academic Press, New York-London.Google Scholar
[40] Knees, D., Rossi, R. & Zanini, C. (2013) A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23, 565–616.Google Scholar
[41] Knees, D., Rossi, R. & Zanini, C. (2015) A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains. Nonlinear Anal. Series B: Real World Appl. 24, 126–162.Google Scholar
[42] Ladyzhenskaya, O. (1991) Attractors for Semigroups and Evolution Equations, Lezioni Lincee, Cambridge University Press, Cambridge.Google Scholar
[43] Laurence, P. & Salsa, S. (2009) Regularity of the free boundary of an American option on several assets. Commun. Pure Appl. Math. 62, 969–994.Google Scholar
[44] Lions, J.-L. & Magenes, E. (1972) Non-Homogeneous Boundary Value Problems and Applications. Vol. I. (Translated from the French by Kenneth, P.), Die Grundlehren der mathematischen Wissenschaften Vol. 181, Springer-Verlag, New York–Heidelberg.Google Scholar
[45] Liu, Q. (2014) Waiting time effect for motion by positive second derivatives and applications. NoDEA Nonlinear Differ. Eq. Appl. 21, 589–620.Google Scholar
[46] Luterotti, F., Schimperna, G. & Stefanelli, U. (2002) Local solution to Frmond's full model for irreversible phase transitions. Mathematical Models and Methods for Smart Materials (Cortona, 2001) Series on Advances in Mathematics for Applied Sciences, Vol. 62, World Scientific Publishing, River Edge, NJ, pp. 323–328.Google Scholar
[47] Rocca, E. & Rossi, R. (2015) “Entropic” solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal. 47, 2519–2586.Google Scholar
[48] Takaishi, T. & Kimura, M. (2009) Phase field model for mode III crack growth in two dimensional elasticity. Kybernetika 45, 605–614.Google Scholar
[49] Voyiadjis, G. Z. & Mozaffari, N. (2013) Nonlocal damage model using the phase field method: Theory and applications. Int. J. Sol. Struct. 50, 3136–3151.Google Scholar
[50] Visintin, A. (1996) Models of Phase Transitions, Progress in Nonlinear Differential Equations and their Applications, Vol. 28. Birkhäuser Boston, Boston, MA.Google Scholar