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Fall 2007 Well-posedness for equations of Benjamin-Ono type
Sebastian Herr
Illinois J. Math. 51(3): 951-976 (Fall 2007). DOI: 10.1215/ijm/1258131113

Abstract

The Cauchy problem $u_t - |D|^{\alpha}u_x + uu_x=0$ in $(-T,T) \times \mathbb{R}$, $u(0)=u_0$, is studied for $1< \alpha <2$. Using suitable spaces of Bourgain type, local well-posedness for initial data $u_0 \in H^s(\mathbb{R}) \cap \dot{H}^{-\omega}(\mathbb{R})$ for any $s > -\tfrac{3}{4}(\alpha-1)$, $\omega:=1/\alpha-1/2$ is shown. This includes existence, uniqueness, persistence, and analytic dependence on the initial data. These results are sharp with respect to the low frequency condition in the sense that if $\omega<1/\alpha-1/2$, then the flow map is not $C^2$ due to the counterexamples previously known. By using a conservation law, these results are extended to global well-posedness in $H^s(\mathbb{R}) \cap \dot{H}^{-\omega}(\mathbb{R})$ for $s \geq 0$, $\omega=1/\alpha-1/2$, and real valued initial data.

Citation

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Sebastian Herr. "Well-posedness for equations of Benjamin-Ono type." Illinois J. Math. 51 (3) 951 - 976, Fall 2007. https://doi.org/10.1215/ijm/1258131113

Information

Published: Fall 2007
First available in Project Euclid: 13 November 2009

zbMATH: 1215.35136
MathSciNet: MR2379733
Digital Object Identifier: 10.1215/ijm/1258131113

Subjects:
Primary: 35Q53
Secondary: 35B30 , 76B03 , 76B15

Rights: Copyright © 2007 University of Illinois at Urbana-Champaign

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Vol.51 • No. 3 • Fall 2007
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