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Solvability Conditions for a Linearized Cahn-Hilliard Equation of Sixth Order

Published online by Cambridge University Press:  29 February 2012

V. Vougalter  and
V. Volpert
V. Vougalter*
Affiliation:
University of Cape Town, Department of Mathematics, Private Bag, Rondebosch 7701, South Africa
V. Volpert
Affiliation:
Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622, France
*
⋆ Corresponding author. E-mail: Vitali.Vougalter@uct.ac.za
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Abstract

We obtain solvability conditions in H6(ℝ3) for a sixth order partial differential equation which is the linearized Cahn-Hilliard problem using the results derived for a Schrödinger type operator without Fredholm property in our preceding article [18].

Keywords

solvability conditionsnon Fredholm operatorsSobolev spaces
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 2: Solitary waves , 2012 , pp. 146 - 154
Copyright
© EDP Sciences, 2012

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References

Alikakos, N.D., Fusco, G.. Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. I. Spectral estimates. Comm. Partial Differential Equations, 19 (1994), No. 9-10, 1397–1447. CrossRefGoogle Scholar
N. Benkirane. Propriété d’indice en théorie Holderienne pour des opérateurs elliptiques dans . CRAS, 307, série I (1988), 577–580.
Caffarelli, L.A., Muler, N.E.. An L∞ bound for solutions of the Cahn-Hilliard equation. Arch. Rational Mech. Anal., 133 (1995), No. 2, 129–144. CrossRefGoogle Scholar
H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon. Schrödinger operators with application to quantum mechanics and global geometry. Springer-Verlag, Berlin, 1987.
Ducrot, A., Marion, M., Volpert, V.. Systemes de réaction-diffusion sans propriété de Fredholm. CRAS, 340 (2005), 659–664. Google Scholar
Ducrot, A., Marion, M., Volpert, V.. Reaction-diffusion problems with non Fredholm operators. Advances Diff. Equations, 13 (2008), No. 11-12, 1151–1192. Google Scholar
Flory, P.J.. Thermodynamics of high polymer solutions. J.Chem.Phys., 10 (1942), 51–61. CrossRefGoogle Scholar
Howard, P.. Spectral analysis of stationary solutions of the Cahn-Hilliard equation. Adv. Differential Equations, 14 (2009), No. 1-2, 87–120. Google Scholar
B.L.G. Jonsson, M. Merkli, I.M. Sigal, F. Ting. Applied Analysis. In preparation.
Kato, T.. Wave operators and similarity for some non-selfadjoint operators. Math. Ann., 162 (1965/1966), 258–279. CrossRefGoogle Scholar
Korzec, M.D., Evans, P.L., Münch, A., Wagner, B.. Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard-type equations. SIAM J. Appl. Math., 69 (2008), No. 2, 348–374. CrossRefGoogle Scholar
E. Lieb, M. Loss. Analysis. Graduate studies in Mathematics, 14. American Mathematical Society, Providence, 1997.
M. Reed, B. Simon. Methods of Modern Mathematical Physics, III : Scattering Theory, Academic Press, 1979.
Rodnianski, I., Schlag, W.. Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math., 155 (2004), No. 3, 451–513. CrossRefGoogle Scholar
Savina, T.V., Golovin, A.A., Davis, S.H., Nepomnyaschy, A.A., oorhees, P.W.V. Faceting of a growing crystal surface by surface diffusion. Phys. Rev. E, 67 (2003), 021606. CrossRefGoogle Scholar
Shchukin, V.A. and Bimberg, D.. Spontaneous ordering of nanostructures on crystal surfaces. Rev. Modern Phys., 71 (1999), No. 4, 1125–1171. CrossRefGoogle Scholar
Volpert, V., Kazmierczak, B., Massot, M., Peradzynski, Z.. Solvability conditions for elliptic problems with non-Fredholm operators. Appl. Math., 29 (2002), No. 2, 219–238. Google Scholar
Vougalter, V., Volpert, V.. Solvability conditions for some non Fredholm operators. Proc. Edinb. Math. Soc. (2), 54 (2011), No. 1, 249–271. CrossRefGoogle Scholar
Vougalter, V., Volpert, V.. On the solvability conditions for some non Fredholm operators. Int. J. Pure Appl. Math., 60 (2010), No. 2, 169–191. Google Scholar
Vougalter, V., Volpert, V.. On the solvability conditions for the diffusion equation with convection terms. Commun. Pure Appl. Anal., 11 (2012), No. 1, 365–373. CrossRefGoogle Scholar
Vougalter, V., Volpert, V.. Solvability relations for some non Fredholm operators. Int. Electron. J. Pure Appl.Math., 2 (2010), No. 1, 75–83. Google Scholar
V. Volpert, V. Vougalter. On the solvability conditions for a linearized Cahn-Hilliard equation. To appear in Rendiconti dell’Instituto di Matematica dell’Universita di Trieste.
Vougalter, V., Volpert, V.. Solvability conditions for some systems with non Fredholm operators. Int. Electron. J. Pure Appl.Math., 2 (2010), No. 3, 183–187. Google Scholar