Modulation equations: Stochastic bifurcation in large domains

Communications in Mathematical Physics

Volumn 258, Issue 2, 2005, Pages 479-512

  Blomker D ^a   Hairer, M ^b   Pavliotis, G A ^b,c  

^a RWTH AACHEN UNIVERSITY   (Germany)

^b UNIVERSITY OF WARWICK   (United Kingdom)

^c IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 22544442514     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-005-1368-8     Document Type: Article

References (22)

1. [Ada75] Adams, R.A.: Sobolev spaces. Pure and Applied Mathematics, Vol. 65, New York-London: Academic Press [A subsidiary of Hurcourt Brace Jovanovich, Publishers], 1975
2. [BH04] Blömker, D.,Hairer, M.: Multiscale expansion of invariant measures for SPDEs. Commun. Math. Phys. 251, 515-555 (2004)
3. [Blö03a] Blömker, D.: Amplitude equations for locally cubic non-autonomous nonlinearities. SIAM J. Appl. Dyn. Syst. 2(2), 464-486 (2003)
4. [Blö03b] Blömker, D.: Approximation of the stochastic Rayleigh-Bénard problem near the onset of instability and related problems, 2003. to appear in Stochastics and Dynamics (SD)
5. [BMPS01] Blömker, D.,Maier-Paape, S.,Schneider, G.: The stochastic Landau equation as an amplitude equation. Discrete and Continuous Dynamical Systems. Series B 1(4), 527-541 (2001)
6. [CE90] Collet, P.,Eckmann, J.-P.: The time dependent amplitude equation for the Swift-Hohenberg problem. Commun. Math. Phys. 132(1), 139-153 (1990)
7. [Cer99] Cerrai, S.: Smoothing properties of transition semigroups relative to SDEs with values in Banach spaces. Probab. Th. Rel. Fields 113(1), 85-114 (1999)
8. [CH93] Cross, M.,Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851-1112 (1993)
9. [DPZ92] Da Prato, G.,Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge: University Press, 1992
10. [DPZ96] Da Prato, G.,Zabczyk, J.: Ergodicity for Infinite Dimensional Systems, Vol. 229 of London Mathematical Society Lecture Note Series. Cambridge: University Press, 1996
11. [EL94] Elworthy, K.D.,Li, X.-M.: Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125(1), 252-286 (1994)
12. [GM01] Goldys, B.,Maslowski, B.: Uniform exponential ergodicity of stochastic dissipative systems. Czech. Math. J. 51(126)(4), 745-762 (2001)
13. [Hai02] Hairer, M.: Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Th. Rel. Fields 124(3), 345-380 (2002)
14. [HS77] Hohenberg, P.,Swift, J.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319-328 (1977)
15. [KSM92] Kirrmann, P.,Schneider, G.,Mielke, A.: The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 122(1-2), 85-91 (1992)
16. [LM99] Lythe, G.,Moro, K.: Dynamics of defect formation. Phys. Rev. E 59, R1303-1306 (1999)
17. [Lyt96] Lythe, G.: Domain formation in transitions with noise and a time-dependent bifurcation parameter. Phys. Rev. E 53, R4271-4274
18. [MS95] Mielke, A.,Schneider, G.: Attractors for modulation equations on unbounded domains - existence and comparison. Nonlinearity 8, 743-768 (1995)
19. [MSZ00] Mielke, A.,Schneider, G.,Ziegra, A.: Comparison of inertial manifolds and application to modulated systems. Math. Nachr. 214, 53-69 (2000)
20. [Rac91] Rachev, S.T.: Probability metrics and the stability of stochastic models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Chichester: John Wiley & Sons Ltd.,1991
21. [Sch96] Schneider, G.: The validity of generalized Ginzburg-Landau equations. Math. Methods Appl. Sci. 19(9), 717-736 (1996)
22. [Sch99] Schneider, G.: Cahn-Hilliard description of secondly flows of a viscous incompressible fluid in an unbounded domain. ZAMM Z. Angew. Math. Mech. 79(9), 615-626 (1999)