Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton

Linares, Felipe; Pastor, Ademir; Saut, Jean‐Claude

doi:10.1080/03605302.2010.494195

Cited by 66 publications

(54 citation statements)

References 14 publications

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“…15) The function on the right-hand side of (3.15) is bounded in L 3/2 (0, T); that is, u u x is bounded in L 3/2 (0, T ; L 9/8 ( )) in dimension 3, (3. 16) and hence it is bounded in L 9/8 (0, T; L 9/8 ( )).…”

Section: B the Existence Of Solutions In Dimension 2 Andmentioning

confidence: 99%

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

Saut¹,

Témam²,

Wang³

2012

Journal of Mathematical Physics

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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...

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Section: B the Existence Of Solutions In Dimension 2 Andmentioning

confidence: 99%

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

Saut¹,

Témam²,

Wang³

2012

Journal of Mathematical Physics

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“…As far as the ZK equation is concerned, there are some recent results [8,10,[20][21][22] on the Cauchy problem and IBVP. Our work was motivated by [29] on IBVP for (1.1) posed on a strip bounded in x variable and unbounded in y.…”

Section: Introductionmentioning

confidence: 99%

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

Larkin¹,

Tronco²

2013

Journal of Differential Equations

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“…Initial-boundary value problems, where y varies in a bounded interval, are less studied, however, from the physical point of view they seem at least the same important ( [19,22,3,18,16,21,5,9,10,11]). For example, there are no results on existence of global solutions in Sobolev spaces with any prescribed regularity.…”

Section: Introduction Description Of Main Resultsmentioning

confidence: 99%

Regular solutions to initial-boundary value problems in a half-strip for two-dimensional Zakharov–Kuznetsov equation

Faminskii

2020

Nonlinear Analysis: Real World Applications

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Initial-boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov-Kuznetsov equation are considered. Results on global well-posedness in classes of regular solutions in the cases of periodic and Neumann boundary conditions, as well as on internal regularity of solutions for all types of boundary conditions are established. Also in the case of Dirichlet boundary conditions one result on long-time decay of regular solutions is obtained.

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Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton

Cited by 66 publications

References 14 publications

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

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

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

Regular solutions to initial-boundary value problems in a half-strip for two-dimensional Zakharov–Kuznetsov equation

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