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

Mathematical Modelling of Natural Phenomena (2012)

- Volume: 7, Issue: 2, page 146-154
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topVougalter, V., and Volpert, V.. "Solvability Conditions for a Linearized Cahn-Hilliard Equation of Sixth Order." Mathematical Modelling of Natural Phenomena 7.2 (2012): 146-154. <http://eudml.org/doc/222314>.

@article{Vougalter2012,

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].},

author = {Vougalter, V., Volpert, V.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {solvability conditions; non Fredholm operators; Sobolev spaces},

language = {eng},

month = {2},

number = {2},

pages = {146-154},

publisher = {EDP Sciences},

title = {Solvability Conditions for a Linearized Cahn-Hilliard Equation of Sixth Order},

url = {http://eudml.org/doc/222314},

volume = {7},

year = {2012},

}

TY - JOUR

AU - Vougalter, V.

AU - Volpert, V.

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

JO - Mathematical Modelling of Natural Phenomena

DA - 2012/2//

PB - EDP Sciences

VL - 7

IS - 2

SP - 146

EP - 154

AB - 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].

LA - eng

KW - solvability conditions; non Fredholm operators; Sobolev spaces

UR - http://eudml.org/doc/222314

ER -

## References

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.