Comparison of inertial manifolds and application to modulated systems

Mathematische Nachrichten

Volumn 214, Issue , 2000, Pages 53-69

  Mielke, A ^a,b   Schneider, G ^a,c   Ziegra, A ^a  

^a Leibniz University Hannover   (Germany)

^b UNIVERSITY OF STUTTGART   (Germany)

^c UNIVERSITY OF BAYREUTH   (Germany)

Author keywords

Invariant manifolds; Pattern formation

Indexed keywords

EID: 0040198048     PISSN: 0025584X     EISSN: None     Source Type: Journal    
DOI: 10.1002/1522-2616(200006)214:1<53::AID-MANA53>3.0.CO;2-4     Document Type: Review

References (23)

1. BRIDGES, T. J., and ROWLANDS, G.: Instability of Spatially Quasi-Periodic States of the Ginzburg-Landau Equations, Proc. Royal Soc. London A 444 (1994), 347-362
2. BRUNOVSKÝ, P.: The Attractor of the Scalar Reaction Diffusion Equation is a Smooth Graph, J. Dynamics Diff. Eqns. 2 (1990), 293-323
3. COLLET, P.: Thermodynamic Limit of the Ginzburg-Landau Equations, Nonlinearity 7 (1994), 1175-1190
4. COLLET, P., and ECKMANN, J.-P.: The Time-Dependent Amplitude Equation for the Swift-Hohenberg Problem, Commun. Math. Phys. 132 (1990), 139-153
5. CONSTANTIN, P., FOIAS, C., NICOLAENKO, B., and TEMAM, R.: Integral and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, 1988
6. DOELMAN, A., GARDNER, R. A., and JONES, C. K. R. T.: Instability of Quasiperiodic Solutions of the Ginzburg-Landau Equation, Proc. Royal Soc. Edinburgh A 125 (1995), 501-517
7. DOERING, C. R., GIBBON, J. D., HOLM, D. D., and NICOLAENKO, B.: Low-Dimensional Behaviour in the Complex Ginzburg-Landau Equation, Nonlinearity 1 (1988), 279-309
8. ECKHAUS, W.: Studies in Non-Linear Stability Theory, Springer tract in Nat. Phil. 6, Springer-Verlag, 1965
9. FOIAS, C., SELL, G. R., and TEMAM, R.: Inertial Manifolds for Nonlinear Evolutionary Equations, J. Diff. Eqns. 73 (1988), 309-353
10. GALLAY, TH.: Periodic Patterns and Traveling Fronts for the Ginzburg-Landau Equation, Proceedings of the IUTAM/ISIMM Symposium on Structure and dynamics of Nonlinear Waves in Fluids held in Hannover 1994, World Scientific, 1995
11. KIRRMANN, P., SCHNEIDER, G., and MIELKE, A.: The Validity of Modulation Equations for Extended Systems with Cubic Nonlinearities, Proc. Royal Soc. Edinburgh 122A (1992), 85-91
12. MIELKE, A.: Essential Manifolds for an Elliptic Problem in an Infinite Strip, J. Diff. Eqns. 110 (1994), 322-355
13. MIELKE, A.: The Complex Ginzburg-Landau Equation on Large and Unbounded Domains: Sharper Bounds and Attractors, Nonlinearity 10 (1997), 199-222
14. MIELKE, A.: Instability and Stability of Rolls in the Swift-Hohenberg Equation, Commun. Math. Physics 189 (1997), 829-853
15. MIELKE, A., and SCHNEIDER, G.: Attractors for Modulation Equations on Unbounded Domains-Existence and Comparison, Nonlinearity 8 (1995), 1-26
16. SCHNEIDER, G.: A New Estimate for the Ginzburg-Landau Approximation on the Real Axis, J. Nonlinear Science 4 (1994), 23-34
17. SCHNEIDER, G.: Error Estimates for the Ginzburg-Landau Approximation, J. Appl. Math. Physics (ZAMP) 45 (1994), 433-457
18. SCHNEIDER, G.: Global Existence via Ginzburg-Landau Formalism and Pseudo-Orbits of Ginzburg-Landau Approximations, Commun. Math. Phys. 164 (1994), 157-179
19. SCHNEIDER, G.: Analyticity of Ginzburg-Landau Modes, J. Diff. Eqns. 121 (1995), 233-257
20. TABOADA, M.: Finite-Dimensional Asymptotic Behavior for the Swift-Hohenberg Model of Convection, Nonlinear Analysis 14 (1990), 43-54
21. TEMAM, R.: Infinite-Dimensional Systems in Mechanics and Physics, Springer-Verlag, Berlin, 1988
22. VAN HARTEN, A.: On the Validity of Ginzburg-Landau's Equation, J. Nonlinear Science 1 (1991), 397-422
23. ZlEGRA, A.: Dynamik in modulierten Systemen, Dissertation, Universität Hannover, November 1997