An initial and boundary-value problem for the Zakharov-Kuznestov equation in a bounded domain

作者：Saut Jean Claude^*; Temam Roger; Wang Chuntian

来源：Journal of Mathematical Physics, 2012, 53(11): 115612.

DOI：10.1063/1.4752102

• Fourier transforms
• group theory
• initial value problems
• nonlinear equations
• wave equations

Motivated by the study of boundary control problems for the Zakharov-Kuznetsov equation, we study in this article the initial and boundary value problem for the ZK (short for Zakharov-Kuznetsov) equation posed in a limited domain Omega = (0, 1)(x) x (-pi/2, pi/2)(d), d = 1, 2. This article is related to Saut and Temam ["An initial boundary-value problem for the Zakharov-Kuznetsov equation," Adv. Differ. Equ. 15(11-12), 1001-1031 (2010)] in which the authors studied the same problem in the band (0,1)(x) x R-d, d = 1, 2, but this article is not a straightforward adaptation of Saut and Temam ["An initial boundary-value problem for the Zakharov-Kuznetsov equation," Adv. Differ. Equ. 15(11-12), 1001-1031 (2010)]; indeed many new issues arise, in particular, for the function spaces, due to the loss of the Fourier transform in the tangential directions (orthogonal to 0x). In this article, after studying a number of suitable function spaces, we show the existence and uniqueness of solutions for the linearized equation using the linear semigroup theory. We then continue with the nonlinear equation with the homogeneous boundary conditions. The case of the full nonlinear equation with nonhomogeneous boundary conditions especially needed for the control problems will be studied elsewhere.

• 出版日期2012-11