Almost global existence and transient self similar decay for Poiseuille flow at criticality for exponentially long times

Physica D: Nonlinear Phenomena

Volumn 185, Issue 3-4, 2003, Pages 209-226

  Schneider, Guido ^a   Uecker, Hannes ^a  

^a UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

Almost global existence; Bloch wave analysis; Nonlinear diffusion equations; Poiseuille flow; Renormalization

Indexed keywords

BIFURCATION (MATHEMATICS); DIFFERENTIAL EQUATIONS; FLOW OF FLUIDS; HAMILTONIANS; NONLINEAR EQUATIONS; REYNOLDS NUMBER; SOLUTIONS;

POISEUILLE FLOW;

FLUID MECHANICS;

EID: 0141955991     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(03)00230-6     Document Type: Article

References (30)

1. P. Bollerman, On the theory of validity of amplitude equations, Thesis, University of Utrecht, 1996.
2. P. Bollerman, G. Schneider, A. van Harten, On the Justification of the Ginzburg-Landau approximation, in: A. Doelman, A. van Harten (Eds.), Nonlinear Dynamics and Pattern Formation in Natural Environment, Pitman Research Notes 335, Longman, New York, 1995, pp. 20-36.
3. Bricmont J., Kupiainen A., Renormalization group and the Ginzburg-Landau equation. Commun. Math. Phys. 150:1992;193-208.
4. Bricmont J., Kupiainen A., Stability of moving fronts in the Ginzburg-Landau equation. Commun. Math. Phys. 159:1994;287-318.
5. Bricmont J., Kupiainen A., Lin G., Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Commun. Pure Appl. Math. 47:1994;893-922.
6. xx-f(u) Proc. Roy. Soc. Edin. A. 116:1990;133-160.
7. P. Chossat, G. Iooss, The Couette-Taylor Problem, Appl. Math. Sciences 102, Springer, Berlin, 1994.
8. Collet P., Eckmann J.-P., Epstein H., Diffusive repair for the Ginsburg-Landau equation. Helv. Phys. Acta. 65:1992;56-92.
9. P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981.
10. Eckmann J.-P., Schneider G., Nonlinear stability of bifurcating front solutions for the Taylor-Couette problem. ZAMM. 80:2000;745-753.
11. Eckmann J.-P., Schneider G., Non-linear stability of modulated fronts for the Swift-Hohenberg equation. Commun. Math. Phys. 225:2002;361-397.
12. Eckhaus W., The Ginzburg-Landau equation is an attractor. J. Nonlin. Sci. 3:1993;329-348.
13. Hocking L.M., Stewartson K., On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. Roy. Soc. Lond. A. 326:1972;289-313.
14. Hocking L.M., The stability of flow in an elliptic pipe with large aspect ratio. Quart. J. Mech. Appl. Math. 30:1977;343-353.
15. Iooss G., Théorie non-linéaire de la stabilité des écoulements laminaires dans le cas de l'échange des stabilités. Arch. Ration. Mech. Anal. 40:1971;166-208.
16. John F., Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscr. Math. 28:1979;235-268.
17. John F., Klainerman S., Almost global existence to nonlinear wave equations in three space dimensions. Commun. Pure Appl. Math. 37:1984;443-455.
18. A. Mielke, The complex Ginzburg-Landau equation in its role as a modulation equation, in: B. Fiedler, G. Iooss, N. Kopell, F. Takens (Eds.), Handbook of Dynamical Systems II, Elsevier, Amsterdam, 2002, pp. 759-834.
19. A. Mielke, G. Schneider, H. Uecker, Stability and diffusive dynamics in extended domains, in: B. Fiedler (Ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001, pp. 563-583.
20. Nehorosev N.N., An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. Russ. Math. Surv. 32(6):1977;1-65.
21. M. Reed, B. Simon, Methods of Modern Mathematical Physics IV, Academic Press, New York, 1978.
22. B. Scarpellini, Stability, Instability, and Direct Integrals, Chapman & Hall/CRC Press, London/Boca Raton, FL, 1999.
23. Schmid P.J., Henningson D.S., Optimal energy density growth in Hagen-Poiseuille flow. J. Fluid Mech. 277:1994;197-225.
24. Schneider G., Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation. Commun. Math. Phys. 178:1996;679-702.
25. Schneider G., Nonlinear stability of Taylor-vortices in infinite cylinders. Arch. Ration. Mech. Anal. 144:1998;121-200.
26. Schneider G., Global existence results for pattern forming processes in infinite cylindrical domains - applications to 3D Navier-Stokes problems. J. Math. Pure Appl. 78:1999;265-312.
27. Snoussi S., Tayachi S., Weissler F.B., Asymptotically self-similar global solutions of a general semilinear heat equation. Math. Ann. 321:2001;131-155.
28. Uecker H., Diffusive stability of rolls in the two-dimensional real and complex Swift-Hohenberg equation. Commun. PDE. 24(11-12):1999;2109-2146.
29. Wayne E.C., Invariant manifolds for parabolic partial differential equations on unbounded domains. Arch. Ration. Mech. Anal. 138:1997;279-306.
30. Weissler F.B., Existence and non-existence of global solutions for a semilinear heat equation. Isr. J. Math. 38:1981;29-40.