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

#### Abstract

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

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

Dates
First available in Project Euclid: 18 December 2012