## Differential and Integral Equations

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

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

#### 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