An initial boundary-value problem for the Zakharov-Kuznetsov equation. (English) Zbl 1219.35253

Saut, Jean-Claude; Temam, Roger

Adv. Differ. Equ. 15, No. 11-12, 1001-1031 (2010).

Summary: 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.

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35A02	Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

Zakharov-Kuznetsov equation; regularity; trace theorem