Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations

Biagioni, H. A.; Linares, Felipe

doi:10.1090/s0002-9947-01-02754-4

Cited by 107 publications

(134 citation statements)

References 21 publications

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“…There is a large literature on the Cauchy problem for (DNLS); see [38,39,19,20,36,3,8,9,42,43,15,11,16,23,24] and references therein. Here we are mainly interested in the results of energy space H 1 (R).…”

Section: Dnls and Mass Critical Nlsmentioning

confidence: 99%

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Potential well theory for the derivative nonlinear Schrödinger equation

Hayashi

2021

Analysis & PDE

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We consider the following nonlinear Schrödinger equation of derivative type:(1)this equation is known as a standard derivative nonlinear Schrödinger equation (DNLS), which is mass critical and completely integrable. The equation (1) can be considered as a generalized equation of DNLS while preserving mass criticality and Hamiltonian structure. For DNLS it is known that if the initial data u 0 ∈ H 1 (R) satisfies the mass condition u 0 2 L 2 < 4π, the corresponding solution is global and bounded. In this paper we first establish the mass condition on (1) for general b ∈ R, which is exactly corresponding to 4π-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass threshold value gives the turning point in the structure of potential well generated by solitons. In particular, our results for DNLS give a characterization of both 4π-mass condition and algebraic solitons.

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Section: Dnls and Mass Critical Nlsmentioning

confidence: 99%

“…This result is considered as a natural extension of 2π-mass condition for (DNLS). 3 From the following energy form…”

Section: Dnls and Mass Critical Nlsmentioning

confidence: 99%

Potential well theory for the derivative nonlinear Schrödinger equation

Hayashi

2021

Analysis & PDE

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“…On the other hand, Molinet, Saut, and Tzvetkov [Molinet et al 2001] proved that the flow map associated with BO, when it exists, fails to be C 2 in any Sobolev space H s ‫,)ޒ(‬ s ∈ ‫.ޒ‬ This result is based on the fact that the dispersive smoothing effects of the linear part of BO are not strong enough to control the low-high frequency interactions appearing in the nonlinearity of (1-1). It was improved by Koch and Tzvetkov [2005], who showed that the flow map fails even to be uniformly continuous in H s ‫)ޒ(‬ for s > 0 (see [Biagioni and Linares 2001] for the same result in the case s < − 1 2 ). As the consequence of those results, one cannot solve the Cauchy problem for the Benjamin-Ono equation by a Picard iterative method implemented on the integral equation associated with (1-1) for initial data in the Sobolev space H s ‫,)ޒ(‬ s ∈ ‫.ޒ‬ In particular, the methods introduced by Bourgain [1993b] and Kenig, Ponce, and Vega [Kenig et al 1993; for the Korteweg-de Vries equation do not apply directly to the Benjamin-Ono equation.…”

Section: Introductionmentioning

confidence: 97%

“…The Benjamin-Ono equation is one of the fundamental equations describing the evolution of weakly nonlinear internal long waves. It has been derived by Benjamin [1967] as an approximate model for long-crested unidirectional waves at the interface of a two-layer system of incompressible inviscid fluids, one being infinitely deep. In nondimensional variables, the initial value problem (IVP) associated with the Benjamin-Ono equation (BO) is…”

Section: Introductionmentioning

confidence: 99%

Untitled

2012

Anal. PDE

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We give a detailed microlocal study of X-ray transforms over geodesic-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator. We compute the principal symbol of both operators and the canonical relation associated to the Fourier integral operator. In two dimensions, for the geodesic transform, we show that there is always a cancellation of singularities to some order, and we give an example where that order is infinite; therefore the normal operator is not microlocally invertible in that case. In the case of three dimensions or higher if the canonical relation is a local canonical graph we show microlocal invertibility of the normal operator. Several examples are also studied.Stefanov is partly supported by NSF grant DMS-0800428. Uhlmann is partly supported by NSF FRG grant 0554571 and a Walker Family Endowed Professorship. MSC2000: 53C65.

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“…The first author [23,24] proved that (1) is small data global well-posed and scattering for s ≥ s c if m + d ≥ 4. Well-posedness of the Cauchy problem for (1) in d = 1 whose ∂(u m ) is replaced by ∂ x (|u| 2 u) is intensively studied by many authors (see, for example, [20,21,31,32,13,3,14,22,34,29] references therein). Presence of derivative causes some difficulties.…”

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confidence: 99%

Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity

Hirayama¹,

Okamoto²

2016

DCDS-A

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We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity (i∂t + ∆)u = ±∂(u m) on R d , d ≥ 1, with random initial data, where ∂ is a first order derivative with respect to the spatial variable, for example a linear combination of ∂ ∂x 1 ,. .. , ∂ ∂x d or |∇| = F −1 [|ξ|F ]. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in H s (R d) with s > max d−1 d sc, sc 2 , sc − d 2(d+1) for d + m ≥ 5, where s is below the scaling critical regularity sc := d 2 − 1 m−1 .

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Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations

Abstract: Abstract. Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in H s (R), s < 1/2. This result implies that best result concerning local well-posedness

Cited by 107 publications

References 21 publications

Potential well theory for the derivative nonlinear Schrödinger equation

Potential well theory for the derivative nonlinear Schrödinger equation

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Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity

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