Differential and Integral Equations

Bilinear estimates with applications to the generalized Benjamin-Ono-Burgers equations

Masanori Otani

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Abstract

In this paper, we deal with the well-posedness issues of the generalized Benjamin-Ono-Burgers (gBOB) equations which are interpolated between the ordinary BOB equation and the KdV-Burgers equation with respect to the dispersive terms. We solve the initial-value problem (IVP) with data below $H ^{-1 /2}$, where $s = -1/2$ is the threshold for the well posedness of the Burgers equation. Our proof is based on the method by L. Molinet and F. Ribaud, which is analogous to that developed by J. Bourgain and C.E. Kenig, G. Ponce, and L. Vega. Interestingly, it is known that such a method cannot be applied to the Benjamin-Ono equation with initial data in $H ^s(\mathbb R)$, $s \in \mathbb R$.

Article information

Source
Differential Integral Equations, Volume 18, Number 12 (2005), 1397-1426.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356059717

Mathematical Reviews number (MathSciNet)
MR2174979

Zentralblatt MATH identifier
1212.35326

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35S10: Initial value problems for pseudodifferential operators 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]

Citation

Otani, Masanori. Bilinear estimates with applications to the generalized Benjamin-Ono-Burgers equations. Differential Integral Equations 18 (2005), no. 12, 1397--1426. https://projecteuclid.org/euclid.die/1356059717


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