Differential and Integral Equations

Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equations as a singular limit of the Kuramoto-Sivashinsky system

C.P. Massarolo and A.F. Pazoto

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Abstract

We consider a coupled system of Kuramoto-Sivashinsky equations depending on a suitable parameter $\nu > 0$ and study its asymptotic behavior for $t$ large, as $\nu\rightarrow 0$. Introducing appropriate boundary conditions we show that the energy of the solutions decays exponentially uniformly with respect to the parameter $\nu$. In the limit, as $\nu\rightarrow 0$, we obtain a coupled system of Korteweg-de Vries equations known to describe strong interactions of two long internal gravity waves in a stratified fluid for which the energy tends to zero exponentially as well. The decay fails when the length of the space interval $L$ lies in a set of critical lengths.

Article information

Source
Differential Integral Equations Volume 22, Number 1/2 (2009), 53-68.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356038554

Mathematical Reviews number (MathSciNet)
MR2483012

Zentralblatt MATH identifier
1240.35473

Subjects
Primary: 93D15: Stabilization of systems by feedback 93B05: Controllability 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Citation

Massarolo, C.P.; Pazoto, A.F. Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equations as a singular limit of the Kuramoto-Sivashinsky system. Differential Integral Equations 22 (2009), no. 1/2, 53--68. https://projecteuclid.org/euclid.die/1356038554.


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