Bifurcating fronts for the Taylor-Couette problem in infinite cylinders

Zeitschrift fur Angewandte Mathematik und Physik

Volumn 50, Issue 1, 1999, Pages 120-151

  Haragus Courcelle, Mariana ^a   Schneider, Guido ^b  

^a UNIVERSITY OF STUTTGART   (Germany)

^b Leibniz University Hannover   (Germany)

Author keywords

Bifurcations; Center manifold theory; Oscillating fronts; Spatial dynamics

Indexed keywords

EID: 0033482571     PISSN: 00442275     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00001491     Document Type: Article

References (47)

1. A. Afendikov, A. Mielke, Bifurcation of Poiseuille flow between parallel plates: Three-dimensional solutions with large spanwise wavelength, Arch. Rat. Mech. Anal. 129 (1995), 101-127.
2. J. Bricmont, A. Kupiainen, Renormalization group and the Ginzburg-Landau equation, Comm. Math. Phys. 150 (1992), 193-208.
3. J. Bricmont, A. Kupiainen, Stability of moving fronts in the Ginzburg-Landau equation, Comm. Math. Phys. 159 (1994), 287-318.
4. P. Chossat, G. Iooss, The Taylor-Couette problem, Appl. Math. Sci. 102, Springer, Berlin 1994.
5. S. N. Chow, J. K. Hale, Methods of Bifurcation Theory, Springer, Berlin 1982.
6. P. Collet, J.-P. Eckmann, The existence of dendritic fronts, Comm. Math. Phys. 107 (1986), 39-92.
7. P. Collet, J.-P. Eckmann, The stability of modulated fronts, Helv. Phys. Acta 60 (1987), 969-991.
8. P. Collet, J.-P. Eckmann, The time dependent amplitude equation for the Swift-Hohenberg problem, Comm. Math. Phys. 132 (1990), 139-153.
9. P. Collet, J.-P. Eckmann, Solutions without phase-slip for the Ginsburg-Landau equation, Comm. Math. Phys. 145 (1992), 345-356.
10. P. Collet, J.-P. Eckmann, H. Epstein, Diffusive repair for the Ginsburg-Landau equation, Helv. Phys. Acta 65 (1992), 56-92.
11. W. Eckhaus, The Ginzburg-Landau equation is an attractor, J. Nonlinear Sci. 3 (1993), 329-348.
12. J.-P. Eckmann, C.E. Wayne, Propagating fronts and the center manifold theorem, Comm. Math. Phys. 136 (1991), 285-307.
13. J.-P. Eckmann, H. Epstein, C.E. Wayne, Normal forms for parabolic partial differential equations, Ann. Inst. Henri Poincare, Phys. Theor. 58(3) (1993), 287-308.
14. J.-P. Eckmann, C.E. Wayne, The non-linear stability of front solutions for parabolic partial differential equations, Comm. Math. Phys. 161 (1994), 323-334.
15. J.-P. Eckmann, C.E. Wayne, P. Wittwer, Geometric stability analysis for periodic solutions of the Swift-Hohenberg equation, Comm. Math. Phys. 190 (1997), 173-211.
16. T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity 7 (1994), 741-764.
17. J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Appl. Math. Sciences 42, Springer, Berlin New York 1983.
18. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math. 840, Springer, Berlin 1981.
19. G. Iooss, Existence et stabilité de la solution périodique secondaire intervenant dans les problèmes d'évolution du type Navier-Stokes, Arch. Rat. Mech. Anal. 47 (1972), 301-329.
20. G. Iooss, A. Mielke, Y. Demay, Theory of steady Ginzburg-Landau equation, in hydrodynamic stability problems, Europ. J. Mech. B/ Fluids 8/3 (1989), 229-268.
21. G. Iooss, A. Mielke, Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders, J. Nonlinear Sci. 1 (1991), 107-146.
22. G. Iooss, K. Kirchgässner, Water waves for small surface tension: an approach via normal form, Proc. Roy. Soc. Edinb. 122A (1992), 267-299.
23. T. Kapitula, On the nonlinear stability of plane waves for the Ginzburg-Landau equation, Comm. Pure. Appl. Math. 47 (1994), 831-841.
24. T. Kato, Perturbation Theory for Linear Operators. Springer, Berlin 1966.
25. K. Kirchgässner, Verzweigungslösungen eines stationären hydrodynamischen Randwert-problems, Habilitation Thesis, Universität Freiburg 1966.
26. K. Kirchgässner, P. Sorger, Branching analysis for the Taylor problem. Quart. J. Appl. Math. 22 (1969), 183-209.
27. K. Kirchgässner, Wave solutions of reversible systems and applications, J. Diff. Equ. 45 (1982), 113-127.
28. K. Kirchgässner, Waves in weakly coupled parabolic systems, Lect. Notes in Math. 1107; Nonlinear Analysis and Optimization. (Ed. C. Vinti), Springer, Berlin 1984, pp. 154-180.
29. K. Kirchgässner, Waves in weakly coupled parabolic systems, Lect. Notes in Math. 1107; Nonlinear Analysis and Optimization. (Ed. C. Vinti), Springer, Berlin 1984, pp. 154-180.
30. K. Kirchgässner, Nonlinear resonant surface waves and homoclinic bifurcation, Adv. Appl. Mech. 26 (1988), 135-181.
31. K. Kirchgässner, On the nonlinear dynamics of travelling fronts, J. Diff. Eqns. 96 (1992), 256-278.
32. K. Kirchgässner, G. Raugel, Stability of fronts for a KPP-system. In: Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, (ed.: G. Dangelmayr, B. Fiedler, K. Kirchgässner, A. Mielke.) Pitman Research Notes in Mathematics Series 352 1996, pp. 147-208.
33. A. Mielke, Steady flows of inviscid fluids under localized perturbations, J. Diff. Equ. 65 (1986), 89-116.
34. A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Meth. Appl. Sci. 10 (1988), 51-66.
35. A. Mielke, Saint Venant's problem and semi-inverse solutions in nonlinear elasticity, Arch. Rat. Mech. Anal. 102 (1988), 205-229.
36. A. Mielke, Normal hyperbolicity of center manifolds and Saint Venant's principle, Arch. Rat. Mech. Anal. 110 (1990), 353-372.
37. A. Mielke, Mathematical analysis of sideband instabilities with application to Raleigh-Bénard convection, J. Nonlinear Sci. 7 (1997), 57-99.
38. S.N. Ovchinnikova, V.I. Iudovich, Analysis of secondary steady flow between rotating cylinders. Prikl. Mat. Mech. 32 (1968), 884-894.
39. D. Sather, Transition Solutions in the Taylor Problem, Arch. Rat. Mech. Anal. 121 (1992), 267-301.
40. G. Schneider, Error estimates for the Ginzburg-Landau approximation, J. Appl. Math. Physics (ZAMP) 45 (1994), 433-457.
41. G. Schneider, Analyticity of Ginzburg-Landau modes, J. Diff. Eqns. 121 (1995), 233-257.
42. G. Schneider, Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation, Comm. Math. Phys. 178 (1996), 679-702.
43. G. Schneider, Nonlinear stability of Taylor vortices in infinite cylinders. Habilitation Thesis, Universität Hannover 1997. Arch. Rat Mech. Anal., accepted 1997.
44. R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam 1984.
45. A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations, Dynamics Reported Vol.2. Teubner, Stuttgart 1989, pp. 89-169.
46. A. Vanderbauwhede, G. Iooss, Centre manifold theory in infinite dimensions, Dynamics Reported Vol. 1, Springer, Berlin 1992, pp. 125-163.
47. A. van Harten, On the validity of Ginzburg-Landau's equation, J. Nonlinear Sci. 1 (1991), 397-422.