Illinois Journal of Mathematics

Well-posedness for equations of Benjamin-Ono type

Sebastian Herr

Full-text: Open access

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.

Article information

Source
Illinois J. Math., Volume 51, Number 3 (2007), 951-976.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258131113

Digital Object Identifier
doi:10.1215/ijm/1258131113

Mathematical Reviews number (MathSciNet)
MR2379733

Zentralblatt MATH identifier
1215.35136

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] 76B03: Existence, uniqueness, and regularity theory [See also 35Q35] 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]

Citation

Herr, Sebastian. Well-posedness for equations of Benjamin-Ono type. Illinois J. Math. 51 (2007), no. 3, 951--976. doi:10.1215/ijm/1258131113. https://projecteuclid.org/euclid.ijm/1258131113


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