The Cauchy problem for the Benjamin–Ono equation in<i>L</i><sup>2</sup>revisited

Molinet, Luc; Pilod, Didier

doi:10.2140/apde.2012.5.365

Cited by 93 publications

(121 citation statements)

References 28 publications

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“…(3) For the one dimensional Benjamin-Ono equation, Tao [25] introduced a gauge transformation which allowed him to establish local and global results in H 1 (R). In the end, it was possible to go all the way to L 2 (R) using this gauge transformation; see [9] and [18]. We do not know if there is such a gauge transformation for the higher dimensional Benjamin-Ono equation.…”

Section: Some Remarks Are In Ordermentioning

confidence: 99%

On a Higher Dimensional Version of the Benjamin--Ono Equation

Hickman¹,

Linares²,

Riaño³

et al. 2019

SIAM J. Math. Anal.

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We consider a higher dimensional version of the Benjamin-Ono equation, ∂tu−R 1 ∆u+u∂x 1 u = 0, where R 1 denotes the Riesz transform with respect to the first coordinate. We first establish sharp space-time estimates for the associated linear equation. These estimates enable us to show that the initial value problem for the nonlinear equation is locally well-posed in L 2 -Sobolev spaces H s (R d ), with s > 5/3 if d = 2 and s > d/2 + 1/2 if d 3. We also provide ill-posedness results.With d = 1, the available local well-posedness theory has been based on compactness methods. Indeed, Molinet, Saut and Tzvetkov [19] proved that the problem cannot be solved in L 2 -Sobolev spaces H s by Picard iteration. We will show that

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Section: Some Remarks Are In Ordermentioning

confidence: 99%

On a Higher Dimensional Version of the Benjamin--Ono Equation

Hickman¹,

Linares²,

Riaño³

et al. 2019

SIAM J. Math. Anal.

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“…This consists in introducing a nonlinear gauge transform of the solution that solved an equation with less bad interactions than the original one. This method was proved to be very efficient to obtain the lowest regularity index for solving canonical equations (see [28], [12], [6], [23] for the BO equation and [11] for dispersive generalized BO equation) but has the disadvantage to behave very bad with respect to perturbation of the equation. The second one consists in improving the dispersive estimates by localizing it on space frequency depending time intervals and then mixing it with classical energy estimates.…”

Section: Introductionmentioning

confidence: 99%

Improvement of the energy method for strongly nonresonant dispersive equations and applications

Molinet

Vento

2015

Anal. PDE

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Abstract. In this paper we propose a new approach to prove the local well-posedness of the Cauchy problem associated with strongly non resonant dispersive equations. As an example we obtain unconditional well-posedness of the Cauchy problem below H 1 for a large class of one-dimensional dispersive equations with a dispersion that is greater or equal to the one of the Benjamin-Ono equation. Since this is done without using a gauge transform, this enables us to prove strong convergence results for solutions of viscous versions of these equations towards the purely dispersive solutions.

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“…Hence t → u(t) coincides with the solution, obtained by Molinet in [12] (cf. also [14]). In [4], we deduce from Theorem 1 that all these solutions are almost periodic in time (cf.…”

Section: Introductionmentioning

confidence: 87%

“…In particular, the BO equation on the torus is globally in time well-posed on the space of L 2 −integrable functions (cf. [12], [14]). For an excellent survey, including the derivation of (1), we refer to the recent article by J.C. Saut [16].…”

Section: Introductionmentioning

confidence: 99%