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November  2013, 33(11&12): 5089-5106. doi: 10.3934/dcds.2013.33.5089

Nonlocal phase-field systems with general potentials

Maurizio Grasselli 1, and  Giulio Schimperna 2,

1. 

Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, I-20133 Milano
2. 

Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1, I-27100 Pavia

Received  October 2011 Revised  July 2012 Published  May 2013

We consider a phase-field model of Caginalp type where the free energy depends on the order parameter $\chi$ in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for $\chi$. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, the first author has established the existence of a finite-dimensional global attractor in the case of a potential defined on $(-1,1)$ and singular at the endpoints. Here we examine both the case of regular potentials as well as the case of physically more relevant singular potentials (e.g., logarithmic). We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional global attractor in the present cases as well.
Keywords: nonlocal operators, uniform regularization properties, well-posedness, Phase-field models, regular and singular potentials, finite dimensional global attractors..
Mathematics Subject Classification: 35B41, 35Q79, 37L30, 80A2.
Citation: Maurizio Grasselli, Giulio Schimperna. Nonlocal phase-field systems with general potentials. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5089-5106. doi: 10.3934/dcds.2013.33.5089
References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[2]

S. Armstrong, S. Brown and J. Han, Numerical analysis for a nonlocal phase field system, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 1-19.

[3]

H. Attouch, "Variational Convergence for Functions and Operators," Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984.

[4]

P. W. Bates and F. Chen, Traveling wave solutions for a nonlocal phase-field system, Interfaces Free Bound., 4 (2002), 227-238. doi: 10.4171/IFB/59.

[5]

P. W. Bates, F. Chen and J. Wang, Global existence and uniqueness of solutions to a nonlocal phase-field system, in "US-Chinese Conference on Differential Equations and Applications" (Eds. P. W. Bates, S.-N. Chow, K. Lu and X. Pan), International Press, Cambridge, MA, (1997), 14-21.

[6]

P. W. Bates and J. Han, The Dirichlet problem for a nonlocal Cahn-Hilliard equation, J. Math. Anal. Appl., 311 (2005), 289-312. doi: 10.1016/j.jmaa.2005.02.041.

[7]

P. W. Bates, J. Han and G. Zhao, On a nonlocal phase-field system, Nonlinear Anal., 64 (2006), 2251-2278. doi: 10.1016/j.na.2005.08.013.

[8]

H. Brezis, "Opérateurs Maximaux Monotones et Sémi-groupes de Contractions dans les Espaces de Hilbert," North-Holland Math. Studies, 5, North-Holland, Amsterdam, 1973.

[9]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[10]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[11]

G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Ann. Phys., 172 (1986), 136-155. doi: 10.1016/0003-4916(86)90022-9.

[12]

G. Caginalp and E. Esenturk, A phase field model with non-local and anisotropic potential, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 311-350. doi: 10.3934/dcdss.2011.4.311.

[13]

X. Chen, G. Caginalp and E. Esenturk, A phase field model with non-local and anisotropic potential, Modelling Simul. Mater. Sci. Eng., 19 (2011), 8, 045006. doi: 10.1088/0965-0393/19/4/045006.

[14]

X. Chen, G. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Arch. Rational Mech. Anal., 202 (2011), 349-372. doi: 10.1007/s00205-011-0429-8.

[15]

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

[16]

E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Łojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21. doi: 10.1016/j.jde.2003.10.026.

[17]

M. Grasselli, Finite-dimensional global attractor for a nonlocal phase-field system, preprint, arXiv:1108.0314, Istit. Lombardo Accad. Sci. Lett. Rend. A, to appear.

[18]

M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67.

[19]

M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277.

[20]

M. Grasselli, H. Petzeltová and G. Schimperna, A nonlocal phase-field system with inertial term, Quart. Appl. Math., 65 (2007), 451-469.

[21]

J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Amer. Math. Soc., Providence, RI, 1988.

[22]

N. J. Koksch and A. J. Milani, "An Introduction to Semiflows," Chapman & Hall/CRC, Boca Raton, FL, 2005.

[23]

P. Krejčí and J. Sprekels, Nonlocal phase-field models for non-isothermal phase transitions and hysteresis, Adv. Math. Sci. Appl., 14 (2004), 593-612.

[24]

P. Krejčí and J. Sprekels, Long time behavior of a singular phase transition model, Discrete Contin. Dyn. Syst., 15 (2006), 1119-1135. doi: 10.3934/dcds.2006.15.1119.

[25]

P. Krejčí, E. Rocca and J. Sprekels, Nonlocal temperature-dependent phase-field models for non-isothermal phase transitions, J. Lond. Math. Soc. (2), 76 (2007), 197-210. doi: 10.1112/jlms/jdm032.

[26]

P. Krejčí, E. Rocca and J. Sprekels, A nonlocal phase-field model with nonconstant specific heat, Interfaces Free Bound., 9 (2007), 285-306. doi: 10.4171/IFB/165.

[27]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralts'eva, "Linear and Quasilinear Equations of Parabolic Type," (Russian). Translated from the Russian by S. Smith. Translations of Mathematical Monographs, 23 American Mathematical Society, Providence, R.I., 1967.

[28]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481.

[29]

E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems, Phys. D, 192 (2004), 279-307. doi: 10.1016/j.physd.2004.01.024.

[30]

G. Schimperna, A. Segatti and S. Zelik, Asymptotic uniform boundedness of energy solutions to the Penrose-Fife model, J. Evol. Equ., 12 (2012), 863-890. doi: 10.1007/s00028-012-0159-x.

[31]

J. Sprekels and 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. doi: 10.1016/S0022-247X(02)00559-0.

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[2]

S. Armstrong, S. Brown and J. Han, Numerical analysis for a nonlocal phase field system, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 1-19.

[3]

H. Attouch, "Variational Convergence for Functions and Operators," Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984.

[4]

P. W. Bates and F. Chen, Traveling wave solutions for a nonlocal phase-field system, Interfaces Free Bound., 4 (2002), 227-238. doi: 10.4171/IFB/59.

[5]

P. W. Bates, F. Chen and J. Wang, Global existence and uniqueness of solutions to a nonlocal phase-field system, in "US-Chinese Conference on Differential Equations and Applications" (Eds. P. W. Bates, S.-N. Chow, K. Lu and X. Pan), International Press, Cambridge, MA, (1997), 14-21.

[6]

P. W. Bates and J. Han, The Dirichlet problem for a nonlocal Cahn-Hilliard equation, J. Math. Anal. Appl., 311 (2005), 289-312. doi: 10.1016/j.jmaa.2005.02.041.

[7]

P. W. Bates, J. Han and G. Zhao, On a nonlocal phase-field system, Nonlinear Anal., 64 (2006), 2251-2278. doi: 10.1016/j.na.2005.08.013.

[8]

H. Brezis, "Opérateurs Maximaux Monotones et Sémi-groupes de Contractions dans les Espaces de Hilbert," North-Holland Math. Studies, 5, North-Holland, Amsterdam, 1973.

[9]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[10]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[11]

G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Ann. Phys., 172 (1986), 136-155. doi: 10.1016/0003-4916(86)90022-9.

[12]

G. Caginalp and E. Esenturk, A phase field model with non-local and anisotropic potential, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 311-350. doi: 10.3934/dcdss.2011.4.311.

[13]

X. Chen, G. Caginalp and E. Esenturk, A phase field model with non-local and anisotropic potential, Modelling Simul. Mater. Sci. Eng., 19 (2011), 8, 045006. doi: 10.1088/0965-0393/19/4/045006.

[14]

X. Chen, G. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Arch. Rational Mech. Anal., 202 (2011), 349-372. doi: 10.1007/s00205-011-0429-8.

[15]

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

[16]

E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Łojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21. doi: 10.1016/j.jde.2003.10.026.

[17]

M. Grasselli, Finite-dimensional global attractor for a nonlocal phase-field system, preprint, arXiv:1108.0314, Istit. Lombardo Accad. Sci. Lett. Rend. A, to appear.

[18]

M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67.

[19]

M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277.

[20]

M. Grasselli, H. Petzeltová and G. Schimperna, A nonlocal phase-field system with inertial term, Quart. Appl. Math., 65 (2007), 451-469.

[21]

J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Amer. Math. Soc., Providence, RI, 1988.

[22]

N. J. Koksch and A. J. Milani, "An Introduction to Semiflows," Chapman & Hall/CRC, Boca Raton, FL, 2005.

[23]

P. Krejčí and J. Sprekels, Nonlocal phase-field models for non-isothermal phase transitions and hysteresis, Adv. Math. Sci. Appl., 14 (2004), 593-612.

[24]

P. Krejčí and J. Sprekels, Long time behavior of a singular phase transition model, Discrete Contin. Dyn. Syst., 15 (2006), 1119-1135. doi: 10.3934/dcds.2006.15.1119.

[25]

P. Krejčí, E. Rocca and J. Sprekels, Nonlocal temperature-dependent phase-field models for non-isothermal phase transitions, J. Lond. Math. Soc. (2), 76 (2007), 197-210. doi: 10.1112/jlms/jdm032.

[26]

P. Krejčí, E. Rocca and J. Sprekels, A nonlocal phase-field model with nonconstant specific heat, Interfaces Free Bound., 9 (2007), 285-306. doi: 10.4171/IFB/165.

[27]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralts'eva, "Linear and Quasilinear Equations of Parabolic Type," (Russian). Translated from the Russian by S. Smith. Translations of Mathematical Monographs, 23 American Mathematical Society, Providence, R.I., 1967.

[28]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481.

[29]

E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems, Phys. D, 192 (2004), 279-307. doi: 10.1016/j.physd.2004.01.024.

[30]

G. Schimperna, A. Segatti and S. Zelik, Asymptotic uniform boundedness of energy solutions to the Penrose-Fife model, J. Evol. Equ., 12 (2012), 863-890. doi: 10.1007/s00028-012-0159-x.

[31]

J. Sprekels and 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. doi: 10.1016/S0022-247X(02)00559-0.

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