# Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation

Abstract : In this paper, we prove that the Cauchy problem associated to the following higher-order Benjamin-Ono equation $\partial_tv-b\mathcal{H}\partial^2_xv- a\epsilon \partial_x^3v=cv\partial_xv-d\epsilon \partial_x(v\mathcal{H}\partial_xv+\mathcal{H}(v\partial_xv)),$ is globally well-posed in the energy space $H^1(\mathbb R)$. Moreover, we study the limit behavior when the small positive parameter $\epsilon$ tends to zero and show that, under a condition on the coefficients $a$, $b$, $c$ and $d$, the solution $v_{\epsilon}$ to this equation converges to the corresponding solution of the Benjamin-Ono equation.
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Journal articles

https://hal.archives-ouvertes.fr/hal-00637981
Contributor : Luc Molinet <>
Submitted on : Thursday, November 3, 2011 - 2:22:44 PM
Last modification on : Wednesday, June 19, 2019 - 1:26:21 AM
Document(s) archivé(s) le : Saturday, February 4, 2012 - 2:25:56 AM

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### Identifiers

• HAL Id : hal-00637981, version 1
• ARXIV : 1111.0858

### Citation

Luc Molinet, Didier Pilod. Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation. Communications in Partial Differential Equations, Taylor & Francis, 2012, 37 (11), pp.2050-2080. ⟨hal-00637981⟩

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