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

Manifold 

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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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