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Application: The Kuramoto—Sivashinsky Equation

  • P. Constantin
  • C. Foias
  • B. Nicolaenko
  • R. Teman
Part of the Applied Mathematical Sciences book series (AMS, volume 70)

Abstract

We recall that in the case of the Kuramoto—Sivashinsky [HN, HN1,HNZ, NSh] equation on the space H of odd L-periodic functions, (du/dt)+Au+R(u)=0, we have
$$R\left( u \right)B\left( {u,u} \right) + Cu + f,$$
(15.1)
with
$$B\left( {u,\upsilon } \right) = u\frac{{d\upsilon }}{{dx}},$$
$$Cu = - {A^{1/2}}u + B(u,\varphi ) + B\left( {\varphi ,u} \right),$$
$$f = A\varphi + \psi {\text{ with}}\psi = \frac{{{d^2}\varphi }}{{d{x^2}}} + \varphi \frac{{d\varphi }}{{dx}},$$
with the explicit time-independent ϕ defined in [FNST, FNST1],
$${\Lambda _n} = {\lambda _n} = {c_0}{\left( {\frac{n}{L}} \right)^4},{\text{ }}n = 1,2,....$$
(Here as in the sequel c0, c1 … denote absolute constants; for instance, c0 = (2π)4.) Also we shall consider L ≥ 1, the case L < 1 being of no interest.

Keywords

Global Attractor Supplementary Condition Absolute Constant Integral Manifold Inertial Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • P. Constantin
    • 1
  • C. Foias
    • 2
  • B. Nicolaenko
    • 3
  • R. Teman
    • 4
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA
  3. 3.Center for Nonlinear StudiesLos Alamos National LaboratoryLos AlamosUSA
  4. 4.Department de MathematiquesUniversité de Paris-SudOrsayFrance

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