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The Cauchy problem for the Ostrovsky equation with positive dispersion

  • Wei Yan
  • Yongsheng Li
  • Jianhua Huang
  • Jinqiao Duan
Article
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Abstract

This paper is devoted to studying the Cauchy problem for the Ostrovsky equation
$$\begin{aligned} \partial _{x}\left( u_{t}-\beta \partial _{x}^{3}u +\frac{1}{2}\partial _{x}(u^{2})\right) -\gamma u=0, \end{aligned}$$
with positive \(\beta \) and \(\gamma \). This equation describes the propagation of surface waves in a rotating oceanic flow. We first prove that the problem is locally well-posed in \(H^{-\frac{3}{4}}(\text{ R })\). Then we reestablish the bilinear estimate, by means of the Strichartz estimates instead of calculus inequalities and Cauchy–Schwartz inequalities. As a byproduct, this bilinear estimate leads to the proof of the local well-posedness of the problem in \(H^{s}(\text{ R })\) for \( s>-\frac{3}{4}\), with help of a fixed point argument.

Keywords

Ostrovsky equation with positive dispersion Cauchy problem Bilinear estimates Strichartz estimates 

Mathematics Subject Classification

Primary 35Q53 Secondary 35B30 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Wei Yan
    • 1
  • Yongsheng Li
    • 2
  • Jianhua Huang
    • 3
  • Jinqiao Duan
    • 4
  1. 1.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangChina
  2. 2.School of MathematicsSouth China University of TechnologyGuangzhouChina
  3. 3.College of ScienceNational University of Defense TechnologyChangshaChina
  4. 4.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA

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