Differential and Integral Equations

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

Stéphane Vento

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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


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