Cauchy problem for the fifth order Kadomtsev-Petviashvili (KPII) equation

Isaza, Pedro; López, J.; Mejía, Jorge

doi:10.3934/cpaa.2006.5.887

Cited by 20 publications

(25 citation statements)

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“…They also proved GWP for initial data in L 2 (R 2 ). Their result was improved by Isaza et al in [17] where LWP was established for s 1 > − 5 4 and s 2 ≥ 0. The authors also employed the almost conservation machinery of the "I-method" to obtain GWP for s 1 > − 4 7 and s 2 ≥ 0.…”

Section: Introductionmentioning

confidence: 92%

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The Fifth Order KP--II Equation on the Upper Half--plane

Erdoğan¹,

Gürel

Tzirakis³

2019

Preprint

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In this paper we study the fifth order Kadomtsev-Petviashvili II (KP-II) equation on the upper half-plane U = {(x, y) ∈ R 2 : y > 0}. In particular we obtain low regularity local well-posedness using the restricted norm method of Bourgain and the Fourier-Laplace method of solving initial and boundary value problems. Moreover we prove that the nonlinear part of the solution is in a smoother space than the initial data.

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Section: Introductionmentioning

confidence: 92%

“…By symmetry, we assume that τ 1 max(M, τ ). Integrating in λ 1 , we estimate the integral in (17) by…”

Section: Noting Thatmentioning

confidence: 99%

The Fifth Order KP--II Equation on the Upper Half--plane

Erdoğan¹,

Gürel

Tzirakis³

2019

Preprint

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“…We may now begin the proof of Theorem 1.2. Since this is a modification of the proof of Isaza, López, and Mejía in [3], we will merely outline the essential steps. To begin, consider the linear problem…”

Section: Preliminariesmentioning

confidence: 99%

“…. To this operator, we apply the following estimate, which follows from equations (1.2) and (1.4) of [3] and the commutativity of Fourier multipliers:…”

Section: Preliminariesmentioning

confidence: 99%

Well-posedness and regularity of the fifth order Kadomtsev–Petviashvili I equation in the analytic Bourgain spaces

et al. 2020

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“…The Cauchy problem for higher order KP equations has been extensively studied (see [5,13,15,20,26,31,32,33,35,42,43] and references therein). For the IVP associated to (4) see e.g.…”

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confidence: 99%

On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping

Moura¹,

Nascimento²,

Santos³

2022

EECT

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In this paper we prove the exponential decay of the energy for the high-order Kadomtsev-Petviashvili II equation with localized damping. To do that, we use the classical dissipation-observability method and a unique continuation principle introduced by Bourgain in [3] here extended for the highorder Kadomtsev-Petviashvili. A similar result is also obtained for the twodimensional Zakharov-Kuznetsov (ZK)equation. The method of proof works better for the ZK equation, so we were led to make some subtle modifications on it to include KP type equations. In fact, to reach a key estimate we use an anisotropic Gagliardo-Nirenberg inequality to drop the y-derivative of the norm.

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Cauchy problem for the fifth order Kadomtsev-Petviashvili (KPII) equation

Cited by 20 publications

References 0 publications

The Fifth Order KP--II Equation on the Upper Half--plane

The Fifth Order KP--II Equation on the Upper Half--plane

Well-posedness and regularity of the fifth order Kadomtsev–Petviashvili I equation in the analytic Bourgain spaces

On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping

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