Well-Posedness Results for the Three-Dimensional Zakharov--Kuznetsov Equation

Ribaud, Francis; Vento, Stéphane

doi:10.1137/110850566

Cited by 68 publications

(61 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…20 We also refer considered in = (0, 1) x × ( − π /2, π /2) d , with d = 1, 2, u = u xx + ⊥ u, ⊥ u = u yy or u yy + u zz . We will use the notations I x = (0, 1) x , I y = ( − π /2, π /2) y and I z = ( − π /2, π /2) z in the sequel.…”

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

View full text Add to dashboard Cite

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...

show abstract

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

View full text Add to dashboard Cite

show abstract

“…Proof Since we have Proposition 3.1, we obtain the desired estimate on two dimensions if we follow the same lines as the proof in the three dimensional case by Ribaud and Vento [21].…”

Section: Proposition 41 (See [21 Proposition 35]) It Holds Thatmentioning

confidence: 55%

“…Proof The Kato type smoothing effects are already given by Faminskii [3] and Ribaud and Vento [21] for the two and three dimensional cases, respectively. Then we only multiply the frequency-uniform decomposition operators k .…”

Section: Proposition 31 It Holds Thatmentioning

confidence: 92%

“…On the other hand, for the Zakharov-Kuznetsov equation (m = 1), the local well-posedness in H s (R 3 ) for s > 9/8 was proved by Linares and Saut [18]. Ribaud and Vento [21] studied the local well-posedness in H s (R 3 ) for s > 1. Molinet and Pilod [20] proved the global wellposedness in H s (R 3 ) for s > 1.…”

Section: The Generalized Zakharov-kuznetsov Equation On Three Dimensionsmentioning

confidence: 99%

See 1 more Smart Citation

Well-Posedness for the Generalized Zakharov–Kuznetsov Equation on Modulation Spaces

Kato

2016

J Fourier Anal Appl

View full text Add to dashboard Cite

We consider the Cauchy problem for the generalized Zakharov-Kuznetzov equation ∂ t u + ∂ x u = ∂ x (u m+1 ) on two or three space dimensions. We mainly study the two dimensional case and give the local well-posedness and the small data global well-posedness in the modulation space M 2,1 (R 2 ) for m ≥ 4. Moreover, for the quartic case (namely, m = 3), the local well-posedness in M 1/4 2,1 (R 2 ) is given. The well-posedness on three dimensions is also considered.

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Doronin

Larkin

2015

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Initial-boundary value problems for the linear 2D Zakharov-Kuznetsov equation posed on bounded rectangles and on a channel-like strip are considered. We suggest a condition on a part of the boundary such that the evolution problem posed on a bounded rectangle has no critical restrictions on its size. The similar results are obtained both for a 2D strip and a 3D groove with no limitations on their width and gauge. Exponential decay rates of solutions for the original problems are established.

show abstract

Well-Posedness Results for the Three-Dimensional Zakharov--Kuznetsov Equation

Cited by 68 publications

References 10 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

Well-Posedness for the Generalized Zakharov–Kuznetsov Equation on Modulation Spaces

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

Contact Info

Product

Resources

About