The Cauchy Problem for the Fractional Kadomtsev--Petviashvili Equations

Linares, Felipe; Pilod, Didier; Saut, Jean‐Claude

doi:10.1137/17m1145379

Cited by 33 publications

(47 citation statements)

References 50 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, the Whitham equation can be viewed as the one‐dimensional restriction of the full dispersion KP equation introduced in to overcome the “bad” behavior of the dispersion relation of the usual KP equations at low frequencies in x (see also the analysis in ). We refer to for a further study of the Cauchy problem and to for the existence of localized solitary waves, “close” to the usual KP I ones in the case of strong surface tension:

\partial_{t} u + {\overset{c}{true}}_{W W} false(\sqrt{ε} false| D^{ε} false| false) {(() 1 + ε \frac{D_{2}^{2}}{D_{1}^{2}})}^{1 / 2} u_{x} + ε \frac{3}{2} u u_{x} = 0,

with

{\overset{c}{true}}_{W W} false(\sqrt{ε} k false) = {(1 + β ε k^{2})}^{\frac{1}{2}} {((), \frac{tanh \sqrt{ε} k}{\sqrt{ε} k})}^{1 / 2},

where

β \geq 0

is a dimensionless coefficient measuring the surface tension effects and

false| D^{ε} false| = \sqrt{D_{1}^{2} + ε D_{2}^{2}}, D_{1} = \frac{1}{i} \partial_{x}, D_{2} = \frac{1}{i} \partial_{y} .

…”

Section: Introductionmentioning

confidence: 99%

On Whitham and Related Equations

et al. 2017

Self Cite

View full text Add to dashboard Cite

Abstract. The aim of this paper is to study, via theoretical analysis and numerical simulations, the dynamics of Whitham and related equations. In particular we establish rigorous bounds between solutions of the Whitham and KdV equations and provide some insights into the dynamics of the Whitham equation in different regimes, some of them being outside the range of validity of the Whitham equation as a water waves model.

show abstract

\partial_{t} u + {\overset{c}{true}}_{W W} false(\sqrt{ε} false| D^{ε} false| false) {(() 1 + ε \frac{D_{2}^{2}}{D_{1}^{2}})}^{1 / 2} u_{x} + ε \frac{3}{2} u u_{x} = 0,

with

{\overset{c}{true}}_{W W} false(\sqrt{ε} k false) = {(1 + β ε k^{2})}^{\frac{1}{2}} {((), \frac{tanh \sqrt{ε} k}{\sqrt{ε} k})}^{1 / 2},

where

β \geq 0

is a dimensionless coefficient measuring the surface tension effects and

false| D^{ε} false| = \sqrt{D_{1}^{2} + ε D_{2}^{2}}, D_{1} = \frac{1}{i} \partial_{x}, D_{2} = \frac{1}{i} \partial_{y} .

…”

Section: Introductionmentioning

confidence: 99%

On Whitham and Related Equations

et al. 2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…Using the same approach as that used in [17], Linares, Pilod and Saut in [18] considered the family of fractional Kadomtsev-Petviashvili equations7) and established the LWP in Y s of the IVP associated to the equations (1.7), with s > 2 − α/4. The main ingredient in the well-posedness analysis in the works [17] and [18] is a Strichartz estimate, similar to that obtained by Kenig and Koenig in [10] and by Kenig in [9].Inspired by the works [17] and [18], in this paper we also study the relation between the amount of dispersion and the size of the Sobolev space in order to have LWP of the family of IVPs (1.1).Following the scheme developed in [18], we also obtain a Strichartz estimate (see Lemma 3.4 below), which, combined with some energy estimates, allows us to prove the LWP of the IVP (1.…”

mentioning

confidence: 59%

“…Following the scheme developed in [18], we also obtain a Strichartz estimate (see Lemma 3.4 below), which, combined with some energy estimates, allows us to prove the LWP of the IVP (1.…”

mentioning

confidence: 99%

The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations

Bustamante¹,

Urrea²,

Mejía³

2019

Communications on Pure &Amp; Applied Analysis

View full text Add to dashboard Cite

In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equationwhere 0 < α ≤ 1, D α x denotes the operator defined through the Fourier transform byand H denotes the Hilbert transform with respect to the variable x, is locally well posed in the Sobolev space2000 Mathematics Subject Classification. 35Q53, 37K05. Key words and phrases. Benjamin Ono equation. 1 in the less dispersive case (α ∈ (0, 1)), and for s > 3/2 − 3α/8 they proved the LWP of the Cauchy problem for the equation (1.6) in the Sobolev space H s (R). Using the same approach as that used in [17], Linares, Pilod and Saut in [18] considered the family of fractional Kadomtsev-Petviashvili equations7) and established the LWP in Y s of the IVP associated to the equations (1.7), with s > 2 − α/4. The main ingredient in the well-posedness analysis in the works [17] and [18] is a Strichartz estimate, similar to that obtained by Kenig and Koenig in [10] and by Kenig in [9].Inspired by the works [17] and [18], in this paper we also study the relation between the amount of dispersion and the size of the Sobolev space in order to have LWP of the family of IVPs (1.1).Following the scheme developed in [18], we also obtain a Strichartz estimate (see Lemma 3.4 below), which, combined with some energy estimates, allows us to prove the LWP of the IVP (1.

show abstract

“…Coming back to rigorous results, the local Cauchy problem (for a larger class of equations including KP-BO and KP-ILW) is studied in [149] and the global existence and scattering of small solutions is proven in [92].…”

Section: Variamentioning

confidence: 99%

Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE

Saut

2019

Fields Institute Communications

Self Cite

View full text Add to dashboard Cite

This survey article is focused on two asymptotic models for internal waves, the Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations that are integrable by inverse scattering techniques (IST). After recalling briefly their (rigorous) derivations we will review old and recent results on the Cauchy problem, comparing those obtained by IST and PDE techniques and also results more connected to the physical origin of the equations. We will consider mainly the Cauchy problem on the whole real line with only a few comments on the periodic case. We will also briefly discuss some close relevant problems in particular the higher order extensions and the two-dimensional (KP like) versions of the BO and ILW equations.Remark 3. The above derivation was performed for purely gravity waves. Surface tension effects result in adding a third order dispersive term in the asymptotic models. One gets for instance the so-called Benjamin equation (see [30,31]):where δ > 0 measures the capillary effects. This equation, which is in some sense close to the KdV equation, is not known to be integrable. Its solitary waves, the existence of which was proven in [30,31] by the degree-theoretic approach, present oscillatory tails. We refer to [145] for the Cauchy problem in L 2 and to [12,21] for further results on the existence and stability of solitary wave solutions and to [46] for numerical simulations.In presence of surface tension, the ILW equation has to be modified in the same way. We are not aware of mathematical results on the resulting equation.Remark 4. The BO equation was fully justified in [219] as a model of long internal waves in a two-fluid system by taking into account the influence of the surface

show abstract

The Cauchy Problem for the Fractional Kadomtsev--Petviashvili Equations

Abstract: The aim of this paper is to prove various ill-posedness and wellposedness results on the Cauchy problem associated to a class of fractional Kadomtsev-Petviashvili (KP) equations including the KP version of the Benjamin-Ono and Intermediate Long Wave equations. Date: May 22, 2017.

Cited by 33 publications

References 50 publications

On Whitham and Related Equations

On Whitham and Related Equations

The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations

Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE

Contact Info

Product

Resources

About