The Cauchy Problem for the Euler–Poisson System and Derivation of the Zakharov–Kuznetsov Equation

Lannes, David; Linares, Felipe; Saut, Jean‐Claude

doi:10.1007/978-1-4614-6348-1_10

Cited by 107 publications

(107 citation statements)

References 35 publications

(78 reference statements)

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“…The model presented in (1.1) is a bi-dimensional generalization of the Korteweg-de Vries (KdV) equation, and was introduced in [29] to describe the propagation of nonlinear ion-acoustic waves in magnetized plasma (for its rigorous derivation we refer to [17]). This model is widely known as the Zakharov-Kuznetsov (ZK) equation and is extensively studied in the literature, see for example [4,5,18,19,21,22,23] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the periodic Zakharov-Kuznetsov equation

Linares¹,

Panthee²,

Robert³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - A

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We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on T 2 . We prove the local well-posedness for given data in H s (T 2 ) whenever s > 5/3. More importantly, we prove that this equation is of quasi-linear type for initial data in any Sobolev space on the torus, in sharp contrast with its semi-linear character in the R 2 and R × T settings.1991 Mathematics Subject Classification. Primary: 35Q53. Secondary: 35B05.

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Section: Introductionmentioning

confidence: 99%

On the periodic Zakharov-Kuznetsov equation

Linares¹,

Panthee²,

Robert³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - A

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“…A rigorous derivation is provided in Ref. 12. Note than c can be eliminated by a change of variable when the problem is posed in the whole space, but not, as it will be the case here, when the spatial domain is limited in the x variable.…”

Section: Introductionmentioning

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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“…We note that K 2 w = β * w and, by (7), K is an invertible bounded operator on H s . We then convert (5) to…”

Section: Preliminariesmentioning

confidence: 99%

Comparison of nonlocal nonlinear wave equations in the long-wave limit

Erbay

Erkip

2019

Applicable Analysis

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We consider a general class of convolution-type nonlocal wave equations modeling bidirectional nonlinear wave propagation. The model involves two small positive parameters measuring the relative strengths of the nonlinear and dispersive effects. We take two different kernel functions that have similar dispersive characteristics in the long-wave limit and compare the corresponding solutions of the Cauchy problems with the same initial data. We prove rigorously that the difference between the two solutions remains small over a long time interval in a suitable Sobolev norm. In particular our results show that, in the long-wave limit, solutions of such nonlocal equations can be well approximated by those of improved Boussinesq-type equations.

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The Cauchy Problem for the Euler–Poisson System and Derivation of the Zakharov–Kuznetsov Equation

Cited by 107 publications

References 35 publications

On the periodic Zakharov-Kuznetsov equation

On the periodic Zakharov-Kuznetsov equation

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

Comparison of nonlocal nonlinear wave equations in the long-wave limit

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