## Differential and Integral Equations

- Differential Integral Equations
- Volume 22, Number 5/6 (2009), 425-446.

### Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity

#### Abstract

We establish the local well posedness of the generalized Benjamin-Ono equation $\partial_tu+\mathcal{H}\partial_x^2u\pm u^k\partial_xu=0$ in $H^s({\mathbb{R}})$, $s > 1/2-1/k$ for $k\geq 12$ and without smallness assumption on the initial data. The condition $s > 1/2-1/k$ is known to be sharp since the solution map $u_0\mapsto u$ is not of class $\mathcal{C}^{k+1}$ on $H^s({\mathbb{R}})$ for $s < 1/2-1/k$. On the other hand, in the particular case of the cubic Benjamin-Ono equation, we prove the ill posedness in $H^s({\mathbb{R}})$, $s < 1/3$.

#### Article information

**Source**

Differential Integral Equations, Volume 22, Number 5/6 (2009), 425-446.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019600

**Mathematical Reviews number (MathSciNet)**

MR2501678

**Zentralblatt MATH identifier**

1240.35494

**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]

#### Citation

Vento, Stéphane. Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity. Differential Integral Equations 22 (2009), no. 5/6, 425--446. https://projecteuclid.org/euclid.die/1356019600