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May/June 2009 Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity
Stéphane Vento
Differential Integral Equations 22(5/6): 425-446 (May/June 2009).

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 $H^s({\mathbb{R}})$, $s < 1/3$.

Citation

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Stéphane Vento. "Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity." Differential Integral Equations 22 (5/6) 425 - 446, May/June 2009.

Information

Published: May/June 2009
First available in Project Euclid: 20 December 2012

zbMATH: 1240.35494
MathSciNet: MR2501678

Subjects:
Primary: 35Q53
Secondary: 35B30

Rights: Copyright © 2009 Khayyam Publishing, Inc.

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Vol.22 • No. 5/6 • May/June 2009
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