Advances in Differential Equations

An initial boundary-value problem for the Zakharov-Kuznetsov equation

Jean-Claude Saut and Roger Temam

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We introduce and study an initial and boundary-value problem for the Zakharov-Kuznetsov equation posed on an infinite strip of ${\mathbb{R}^{d+1}}$, $d=1,2$. After establishing a suitable trace theorem, we first consider the linearized case and define the corresponding semigroup on $L^2$ and prove that it has a global smoothing effect. Then we proceed to the nonlinear case and use the smoothing effect to prove in both dimensions the existence of (unique when $d=1$) global weak solutions of the initial and boundary problem with null boundary conditions and $L^2$ initial data.

Article information

Adv. Differential Equations, Volume 15, Number 11/12 (2010), 1001-1031.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35A01: Existence problems: global existence, local existence, non-existence 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness


Saut, Jean-Claude; Temam, Roger. An initial boundary-value problem for the Zakharov-Kuznetsov equation. Adv. Differential Equations 15 (2010), no. 11/12, 1001--1031.

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