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

Abstract

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 = (0, 1) x × ( − π /2, π /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 × 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. C 2012 American Institute of Physics. to Ref. 15 for solutions on a nontrivial background and to Ref. 21 for well-posedness results of the Cauchy problem for generalized Zakharov-Kuznetsov equations in R 2 .For both numerical purposes and problems of controllability (see, for instance, Ref. 22), many articles have been recently devoted to the study of initial-value boundary problems for nonlinear dispersive equations, especially in one spatial dimension. We refer, for instance, to Refs. 1, 3, and 4 for the Korteweg-de Vries equation posed on a bounded interval (0, L).Concerning the initial-value boundary problem associated with (1.1), the initial-value boundary problem has been studied in the half space {(x, y): x > 0}, 6 and on a strip 24 We also mention Ref. 13, where the damping effect of boundary conditions is used to study the time rate of decay of small solutions.The initial-boundary value problem (IBVP) for the Zakharov-Kuznetsov equation in a rectangle or in a strip with nonhomogeneous boundary data is studied in Ref. 19, together with the associated boundary control. It is in particular proven in Ref. 19 by a contraction argument that the IBVP is locally well-posed for small data in L 2 ( ), where = (0, L)×T or (0, L)×T 2 , a result which is needed for controllability on the right or on the left.We consider here arbitrary large data. The present article can be viewed as an extension to a bounded domain of the results of Ref. 24 which has focussed on the case of a strip and our method is based on compactness. We consider here a limited domain {(x, x ⊥ ), 0 < x < 1, x ⊥ ∈ ( − π /2, π /2) d , d = 1, 2}, in which case we need to introduce boundary conditions on the y and z boundaries. Here, the boundary conditions that we consider in the y...

Regular solutions of the 2D Zakharov–Kuznetsov equation on a half-strip

“…To prove the existence part of this theorem, we put α = 0 and use as in [29] a parabolic regularization of (2.1)-(2.3), that is, we consider for any (small) positive > 0 the following parabolic problem:…”

“…Our work was motivated by [29] on IBVP for (1.1) posed on a strip bounded in x variable and unbounded in y. Studying this paper, we have found that the term u xyy in (1.1) delivers additional "dissipation" which helped to prove decay of small solutions in domains of a channel type unbounded in x direction.…”

“…More precisely, we consider the IBVP (2.1)-(2.3). In order to demonstrate existence of global regular solutions, we exploit as in [29] a parabolic regularization (3.1)-(3.3). Estimates, independent of a parameter of regularization , permit us to establish existence of regular solutions for the original problem (2.1)-(2.3).…”

Decay of Small Solutions for the Zakharov-Kuznetsov Equation posed on a half-strip

“…Our work was motivated by the paper of Saut and Temam [22] on initial boundary value problem in a domain bounded in x variable and non-bounded in y variable. Studying this paper, we discovered that the term u xyy in ZK equation delivers additional "dissipation" which can help to prove decay of small solutions in non-bounded domains of a channel type non-bounded in x direction and we consider the following initial boundary value problem.…”

Stabilization for the linear Zakharov–Kuznetsov equation without critical size restrictions

“…Quite recently, the interest on dispersive equations became to be extended to multi-dimensional models such as the Kadomtsev-Petviashvili (KP) and ZK equations. As far as the ZK equation is concerned, the results on IVP and IBVP can be found in [8,9,11,21,23,15,18,26,22,31,32]. Our work has been inspired by [30] where (1.2) posed on a strip bounded in x variable was concerned with.…”