Skip to main content
SpringerLink
Log in

A global attracting set for the Kuramoto-Sivashinsky equation

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

New bounds are given for the L2-norm of the solution of the Kuramoto-Sivashinsky equation

$$\partial _t U(x,t) = - (\partial _x^2 + \partial _x^4 )U(x,t) - U(x,t)\partial _x U(x,t)$$

, for initial data which are periodic with periodL. There is no requirement on the antisymmetry of the initial data. The result is

$$\mathop {\lim \sup }\limits_{t \to \infty } \left\| {U( \cdot ,t)} \right\|_2 \leqslant const. L^{8/5} $$

.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • [Felen] Frisch, U., She, Z. S., Thual, O.: Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model. J. Fluid Mech.168, 221–240 (1986)

    Google Scholar 

  • [Il] Ilyashenko, Yu.S.: Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation. J. Dyn. Differ. Equations, in print

  • [M] Manneville, P.: Dissipative Structures and Weak Turbulence, San Francisco-London, Academic Press, 1989

    Google Scholar 

  • [Nelen] Nicolaenko, B., Scheurer, B., Temam, R.: Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear, stability and attractors. PhysicaD16, 155–183 (1985)

    Google Scholar 

  • [T] Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Berlin, Heidelberg, New York: Springer, 1988

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS UPR14, Physique Théorique, Ecole Polytechnique, Palaiseau

    Pierre Collet

  2. Dept. de Physique Théorique, Université de Genève, Genève, Italy

    Jean-Pierre Eckmann & Joachim Stubbe

  3. Section de Mathématiques, Université de Genève, Genève, Italy

    Jean-Pierre Eckmann

  4. IHES, Bures-sur-Yvette

    Henri Epstein

Authors
  1. Pierre Collet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Pierre Eckmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Henri Epstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Joachim Stubbe
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

Rights and permissions

Reprints and permissions

About this article

Cite this article

Collet, P., Eckmann, JP., Epstein, H. et al. A global attracting set for the Kuramoto-Sivashinsky equation. Commun.Math. Phys. 152, 203–214 (1993). https://doi.org/10.1007/BF02097064

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02097064

Keywords

Search

Navigation