June 2018, 7(2): 217-245. doi: 10.3934/eect.2018011

Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics

Dipartimento di Matematica, Universitá di Pavia, Via Ferrata 5, Pavia, PV 27100, Italy

* Corresponding author: Michele Colturato

Received  November 2017 Revised  December 2017 Published  May 2018

We consider a singular phase field system located in a smooth bounded domain. In the entropy balance equation appears a logarithmic nonlinearity. The second equation of the system, deduced from a balance law for the microscopic forces that are responsible for the phase transition process, is perturbed by an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. We prove existence and uniqueness of the solution for this resulting highly nonlinear system. Moreover, under further assumptions, the longtime behavior of the solution is investigated.

Citation: Michele Colturato. Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics. Evolution Equations & Control Theory, 2018, 7 (2) : 217-245. doi: 10.3934/eect.2018011
References:
[1]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

[2]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[3]

V. BarbuP. ColliG. GilardiG. Marinoschi and E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim., 55 (2017), 2108-2133. doi: 10.1137/15M102424X.

[4]

V. BarbuP. ColliG. Gilardi and M. Grasselli, Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation, Differential Integral Equations, 13 (2000), 1233-1262.

[5]

J. F. Blowey and C. M. Elliott, A phase-field model with double obstacle potential, Motions by Mean Curvature and Related Topics, De Gruyter, Berlin, 1994, 1–22.

[6]

E. BonettiP. ColliM. Fabrizio and G. Gilardi, Global solution to a singular integrodifferential system related to the entropy balance, Nonlinear Anal., 66 (2007), 1949-1979. doi: 10.1016/j.na.2006.02.035.

[7]

E. BonettiP. ColliM. Fabrizio and G. Gilardi, Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1001-1026. doi: 10.3934/dcdsb.2006.6.1001.

[8]

E. BonettiP. Colli and M. Frémond, A phase field model with thermal memory governed by the entropy balance, Math. Models Methods Appl. Sci., 13 (2003), 1565-1588. doi: 10.1142/S0218202503003033.

[9]

E. BonettiP. Colli and G. Gilardi, Singular limit of an integrodifferential system related to the entropy balance, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1935-1953. doi: 10.3934/dcdsb.2014.19.1935.

[10]

E. Bonetti and M. Frémond, A phase transition model with the entropy balance, Math. Methods Appl. Sci., 26 (2003), 539-556. doi: 10.1002/mma.366.

[11]

E. BonettiM. Frémond and E. Rocca, A new dual approach for a class of phase transitions with memory: existence and long-time behaviour of solutions, J. Math. Pures Appl.(9), 88 (2007), 455-481. doi: 10.1016/j.matpur.2007.09.005.

[12]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.

[13]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[14]

P. Colli and M. Colturato, Global existence for a singular phase field system related to a sliding mode control problem, Appl. Math., 61 (2016), 623–650, arXiv: 1609.00127. doi: 10.1007/s10492-016-0150-x.

[15]

P. Colli and P. Laurençot, Weak solutions to the Penrose-Fife phase field model for a class of admissible heat flux laws, Phys. D, 111 (1998), 311-334. doi: 10.1016/S0167-2789(97)80018-8.

[16]

P. Colli, P. Laurençot and J. Sprekels, Global solution to the Penrose-Fife phase field model with special heat flux laws, in Variations of Domain and Free-Boundary Problems in Solid Mechanics, Solid Mech. Appl., 66, Kluwer Acad. Publ., Dordrecht, 1999,181–188. doi: 10.1007/978-94-011-4738-5_21.

[17]

P. Colli and J. Sprekels, Global solution to the Penrose-Fife phase-field model with zero interfacial energy and Fourier law, Adv. Math. Sci. Appl., 9 (1999), 383-391.

[18]

M. Colturato, Solvability of a class of phase field systems related to a sliding mode control problem, Appl. Math., 61 (2016), 623-650. doi: 10.1007/s10492-016-0150-x.

[19]

M. Colturato, On a class of conserved phase field systems with a maximal monotone perturbation, Appl. Math. Optim., (2017), (see also preprint arXiv: 1609.00127, [math. AP] (2016), 1–35). doi: 10.1007/s00245-017-9415-3.

[20]

M. Fabrizio, Free energies in the materials with fading memory and applications to PDEs, WASCOM 2003–12th Conference on Waves and Stability in Continuous Media, World Sci. Publishing, (2004), 172–184. doi: 10.1142/9789812702937_0022.

[21]

M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.

[22]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.

[23]

A. Haraux, Systémes Dinamiques Dissipatif et Applications, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 17. Masson, Paris, 1991.

[24]

J. W. Jerome, Approximations of Nonlinear Evolution Systems, vol. 164 of Mathematics in Science and Engineering, Academic Press Inc., Orlando, 1983.

[25]

C. Kittel, Introduction to Solid State Physics, John Wiley and Sons, New York, 1966.

[26]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase field type for the kinetics of phase transitions, Physica D, 69 (1993), 107-113.

[27]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl.(4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

show all references

References:
[1]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

[2]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[3]

V. BarbuP. ColliG. GilardiG. Marinoschi and E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim., 55 (2017), 2108-2133. doi: 10.1137/15M102424X.

[4]

V. BarbuP. ColliG. Gilardi and M. Grasselli, Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation, Differential Integral Equations, 13 (2000), 1233-1262.

[5]

J. F. Blowey and C. M. Elliott, A phase-field model with double obstacle potential, Motions by Mean Curvature and Related Topics, De Gruyter, Berlin, 1994, 1–22.

[6]

E. BonettiP. ColliM. Fabrizio and G. Gilardi, Global solution to a singular integrodifferential system related to the entropy balance, Nonlinear Anal., 66 (2007), 1949-1979. doi: 10.1016/j.na.2006.02.035.

[7]

E. BonettiP. ColliM. Fabrizio and G. Gilardi, Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1001-1026. doi: 10.3934/dcdsb.2006.6.1001.

[8]

E. BonettiP. Colli and M. Frémond, A phase field model with thermal memory governed by the entropy balance, Math. Models Methods Appl. Sci., 13 (2003), 1565-1588. doi: 10.1142/S0218202503003033.

[9]

E. BonettiP. Colli and G. Gilardi, Singular limit of an integrodifferential system related to the entropy balance, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1935-1953. doi: 10.3934/dcdsb.2014.19.1935.

[10]

E. Bonetti and M. Frémond, A phase transition model with the entropy balance, Math. Methods Appl. Sci., 26 (2003), 539-556. doi: 10.1002/mma.366.

[11]

E. BonettiM. Frémond and E. Rocca, A new dual approach for a class of phase transitions with memory: existence and long-time behaviour of solutions, J. Math. Pures Appl.(9), 88 (2007), 455-481. doi: 10.1016/j.matpur.2007.09.005.

[12]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.

[13]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[14]

P. Colli and M. Colturato, Global existence for a singular phase field system related to a sliding mode control problem, Appl. Math., 61 (2016), 623–650, arXiv: 1609.00127. doi: 10.1007/s10492-016-0150-x.

[15]

P. Colli and P. Laurençot, Weak solutions to the Penrose-Fife phase field model for a class of admissible heat flux laws, Phys. D, 111 (1998), 311-334. doi: 10.1016/S0167-2789(97)80018-8.

[16]

P. Colli, P. Laurençot and J. Sprekels, Global solution to the Penrose-Fife phase field model with special heat flux laws, in Variations of Domain and Free-Boundary Problems in Solid Mechanics, Solid Mech. Appl., 66, Kluwer Acad. Publ., Dordrecht, 1999,181–188. doi: 10.1007/978-94-011-4738-5_21.

[17]

P. Colli and J. Sprekels, Global solution to the Penrose-Fife phase-field model with zero interfacial energy and Fourier law, Adv. Math. Sci. Appl., 9 (1999), 383-391.

[18]

M. Colturato, Solvability of a class of phase field systems related to a sliding mode control problem, Appl. Math., 61 (2016), 623-650. doi: 10.1007/s10492-016-0150-x.

[19]

M. Colturato, On a class of conserved phase field systems with a maximal monotone perturbation, Appl. Math. Optim., (2017), (see also preprint arXiv: 1609.00127, [math. AP] (2016), 1–35). doi: 10.1007/s00245-017-9415-3.

[20]

M. Fabrizio, Free energies in the materials with fading memory and applications to PDEs, WASCOM 2003–12th Conference on Waves and Stability in Continuous Media, World Sci. Publishing, (2004), 172–184. doi: 10.1142/9789812702937_0022.

[21]

M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.

[22]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.

[23]

A. Haraux, Systémes Dinamiques Dissipatif et Applications, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 17. Masson, Paris, 1991.

[24]

J. W. Jerome, Approximations of Nonlinear Evolution Systems, vol. 164 of Mathematics in Science and Engineering, Academic Press Inc., Orlando, 1983.

[25]

C. Kittel, Introduction to Solid State Physics, John Wiley and Sons, New York, 1966.

[26]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase field type for the kinetics of phase transitions, Physica D, 69 (1993), 107-113.

[27]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl.(4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

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