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August  2013, 18(6): 1581-1610. doi: 10.3934/dcdsb.2013.18.1581

Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions

Ciprian G. Gal 1, and  Maurizio Grasselli 2,

1. 

Department of Mathematics, Florida International University, Miami, FL, 33199, United States
2. 

Dipartimento di Matematica, Politecnico di Milano, 20133 Milano

Received  October 2011 Revised  December 2011 Published  March 2013

We consider a modified Cahn-Hiliard equation where the velocity of the order parameter $u$ depends on the past history of $\Delta \mu $, $\mu $ being the chemical potential with an additional viscous term $ \alpha u_{t},$ $\alpha >0.$ In addition, the usual no-flux boundary condition for $u$ is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The aim of this work is to analyze the passage to the singular limit when the memory kernel collapses into a Dirac mass. In particular, we discuss the convergence of solutions on finite time-intervals and we also establish stability results for global and exponential attractors.
Keywords: robust exponential attractors, memory relaxation., dynamic boundary conditions, Cahn-Hilliard equations, global attractors.
Mathematics Subject Classification: 35B41, 35Q99, 35R09, 37L30, 82C2.
Citation: Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581
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show all references

References:
[1]

NoDEA Nonlinear Differential Equations Appl., 17 (2010), 663-695. doi: 10.1007/s00030-010-0075-0.  Google Scholar

[2]

J. Chem. Phys., 28 (1958), 258-267. Google Scholar

[3]

Nonlinear Anal., 72 (2010), 2375-2399. doi: 10.1016/j.na.2009.11.002.  Google Scholar

[4]

Asymptot. Anal., 71 (2011), 123-162.  Google Scholar

[5]

American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[6]

Math. Nachr., 13 (2006), 1448-1462. doi: 10.1002/mana.200410431.  Google Scholar

[7]

Nonlinear Anal., 72 (2010), 1668-1682. doi: 10.1016/j.na.2009.09.006.  Google Scholar

[8]

Quart. Appl. Math., 68 (2010), 607-643.  Google Scholar

[9]

Math. Meth. Appl. Sci., 32 (2009), 1370-1395. doi: 10.1002/mma.1091.  Google Scholar

[10]

Indiana Univ. Math. J., 55 (2006), 169-215. doi: 10.1512/iumj.2006.55.2661.  Google Scholar

[11]

Arch. Rational Mech. Anal., 37 (1970), 297-308.  Google Scholar

[12]

Ann. Rev. Fluid Mech., 11 (1979), 371-400. Google Scholar

[13]

Math. Nachr., 272 (2004), 11-31. doi: 10.1002/mana.200310186.  Google Scholar

[14]

Phys. Rev. Letters, 79 (1997), 893-896. Google Scholar

[15]

Europhys. Letters, 42 (1998), 49-54. Google Scholar

[16]

Math. Methods Appl. Sci., 29 (2006), 2009-2036. doi: 10.1002/mma.757.  Google Scholar

[17]

Dyn. Partial Differ. Equ., 5 (2008), 39-67.  Google Scholar

[18]

Commun. Pure Appl. Anal., 7 (2008), 819-836. doi: 10.3934/cpaa.2008.7.819.  Google Scholar

[19]

C. G. Gal, Exponential attractors for a Cahn-Hilliard model in bounded domains with permeable walls,, Electron. J. Differential Equations, 2006 ().   Google Scholar

[20]

C. G. Gal, G. R. Goldstein, J. A. Goldstein, S. Romanelli and M. Warma, Fredholm alternative, semilinear elliptic problems, and Wentzell boundary conditions,, submitted., ().   Google Scholar

[21]

Nonlinear Anal. Real World Appl., 10 (2009), 1738-1766. doi: 10.1016/j.nonrwa.2008.02.013.  Google Scholar

[22]

Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 113-147. doi: 10.3934/dcdss.2009.2.113.  Google Scholar

[23]

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[24]

Phys. Rev. E, 71 (2005), 046125, 13 pp. Google Scholar

[25]

Phys. A, 388 (2009), 3113-3123. doi: 10.1016/j.physa.2009.04.003.  Google Scholar

[26]

Philos. Mag. Lett., 87 (2007), 821-827. Google Scholar

[27]

Int. J. Thermodyn., 11 (2008), 21-28. Google Scholar

[28]

Phys. Lett. A, 372 (2008), 985-989. Google Scholar

[29]

Math. Models Methods Appl. Sci., 15 (2005), 165-198. doi: 10.1142/S0218202505000327.  Google Scholar

[30]

Nonlinearity, 18 (2005), 1859-1883. doi: 10.1088/0951-7715/18/4/023.  Google Scholar

[31]

in "Dissipative Phase Transitions," Ser. Adv. Math. Appl. Sci., 71, World Sci. Publ., Hackensack, NJ, (2006), 101-114. doi: 10.1142/9789812774293_0006.  Google Scholar

[32]

Proc. Royal Soc. Edinburgh Sect. A, 140 (2010), 329-366. doi: 10.1017/S0308210509000365.  Google Scholar

[33]

Rend. Cl. Sci. Mat. Nat., 141 (2007), 129-161.  Google Scholar

[34]

Comm. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.  Google Scholar

[35]

Phys. D, 240 (2011), 754-766. doi: 10.1016/j.physd.2010.12.007.  Google Scholar

[36]

Asymptot. Anal., 56 (2008), 229-249.  Google Scholar

[37]

Asymptot. Anal., 33 (2003), 261-320.  Google Scholar

[38]

J. Evol. Equ., 9 (2009), 371-404. doi: 10.1007/s00028-009-0017-7.  Google Scholar

[39]

Comm. Partial Differential Equations, 34 (2009), 137-170. doi: 10.1080/03605300802608247.  Google Scholar

[40]

Nonlinearity, 23 (2010), 707-737. doi: 10.1088/0951-7715/23/3/016.  Google Scholar

[41]

in "Evolution Equations, Semigroups and Functional Analysis" (eds. A. Lorenzi and B. Ruf) (Milano, 2000), Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Basel, (2002), 155-178.  Google Scholar

[42]

J. Differential Equations, 239 (2007), 38-60. doi: 10.1016/j.jde.2007.05.003.  Google Scholar

[43]

Phys. D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[44]

Colloq. Math., 109 (2007), 217-229. doi: 10.4064/cm109-2-4.  Google Scholar

[45]

Topol. Methods Nonlinear Anal., 32 (2008), 327-345.  Google Scholar

[46]

Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[47]

Computer Phys. Comm., 133 (2001), 139-157. doi: 10.1016/S0010-4655(00)00159-4.  Google Scholar

[48]

Eur. Phys. J. Special Topics, 177 (2009), 165-175. Google Scholar

[49]

J. Evol. Equ., 7 (2007), 59-78. doi: 10.1007/s00028-006-0235-1.  Google Scholar

[50]

Math. Models Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.  Google Scholar

[51]

in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[52]

Discrete Contin. Dyn. Syst., 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.  Google Scholar

[53]

in "Material Instabilities in Continuum Mechanics" (Edinburgh, 1985-1986), Oxford Sci. Publ., Oxford Univ. Press, New York, (1988), 329-342.  Google Scholar

[54]

in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 201-228. doi: 10.1016/S1874-5717(08)00004-2.  Google Scholar

[55]

Commun. Pure Appl. Anal., 5 (2006), 609-614.  Google Scholar

[56]

Adv. Math. Sci. Appl., 11 (2001), 505-529.  Google Scholar

[57]

Ann. Mat. Pura Appl. (4), 185 (2006), 627-648. doi: 10.1007/s10231-005-0175-3.  Google Scholar

[58]

Comm. Comp. Phys., 1 (2006), 1-52. Google Scholar

[59]

in "Nonlinear Partial Differential Equations and their Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkotōsho, Tokyo, (2004), 266-276.  Google Scholar

[60]

Adv. Differential Equations, 8 (2003), 83-110.  Google Scholar

[61]

Comm. Partial Differential Equations, 24 (1999), 1055-1077. doi: 10.1080/03605309908821458.  Google Scholar

[62]

Math. Models Methods Appl. Sci., 17 (2007), 411-437. doi: 10.1142/S0218202507001978.  Google Scholar

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Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar

[64]

J. Math. Anal. Appl., 328 (2007), 789-812. doi: 10.1016/j.jmaa.2006.05.075.  Google Scholar

[65]

J. Differential Equations, 204 (2004), 511-531. doi: 10.1016/j.jde.2004.05.004.  Google Scholar

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