Differential and Integral Equations

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

Stéphane Vento

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.


Export citation