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

V. Vougalter; V. Volpert

Mathematical Modelling of Natural Phenomena (2012)

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

Abstract

top
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].

How to cite

top

Vougalter, 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
  1. 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.  
  2. N. Benkirane. Propriété d’indice en théorie Holderienne pour des opérateurs elliptiques dans . CRAS, 307, série I (1988), 577–580.  
  3. 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.  
  4. H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon. Schrödinger operators with application to quantum mechanics and global geometry. Springer-Verlag, Berlin, 1987.  
  5. A. Ducrot, M. Marion, V. Volpert. Systemes de réaction-diffusion sans propriété de Fredholm. CRAS, 340 (2005), 659–664.  
  6. A. Ducrot, M. Marion, V. Volpert. Reaction-diffusion problems with non Fredholm operators. Advances Diff. Equations, 13 (2008), No. 11-12, 1151–1192.  
  7. P.J. Flory. Thermodynamics of high polymer solutions. J.Chem.Phys., 10 (1942), 51–61.  
  8. P. Howard. Spectral analysis of stationary solutions of the Cahn-Hilliard equation. Adv. Differential Equations, 14 (2009), No. 1-2, 87–120.  
  9. B.L.G. Jonsson, M. Merkli, I.M. Sigal, F. Ting. Applied Analysis. In preparation.  
  10. T. Kato. Wave operators and similarity for some non-selfadjoint operators. Math. Ann., 162 (1965/1966), 258–279.  
  11. 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.  
  12. E. Lieb, M. Loss. Analysis. Graduate studies in Mathematics, 14. American Mathematical Society, Providence, 1997.  
  13. M. Reed, B. Simon. Methods of Modern Mathematical Physics, III : Scattering Theory, Academic Press, 1979.  
  14. 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.  
  15. 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.  
  16. V.A. Shchukin and D. Bimberg. Spontaneous ordering of nanostructures on crystal surfaces. Rev. Modern Phys., 71 (1999), No. 4, 1125–1171.  
  17. 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.  
  18. V. Vougalter, V. Volpert. Solvability conditions for some non Fredholm operators. Proc. Edinb. Math. Soc. (2), 54 (2011), No. 1, 249–271.  
  19. V. Vougalter, V. Volpert. On the solvability conditions for some non Fredholm operators. Int. J. Pure Appl. Math., 60 (2010), No. 2, 169–191.  
  20. 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.  
  21. V. Vougalter, V. Volpert. Solvability relations for some non Fredholm operators. Int. Electron. J. Pure Appl.Math., 2 (2010), No. 1, 75–83.  
  22. 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.  
  23. 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 ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.