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Low Regularity for the Higher Order Nonlinear Dispersive Equation in Sobolev Spaces of Negative Index

  • Zaiyun ZhangEmail author
  • Zhenhai Liu
  • Mingbao Sun
  • Songhua Li
Article
  • 106 Downloads

Abstract

In this paper, we investigate the initial value problem(IVP henceforth) associated with the higher order nonlinear dispersive equation given in Jones et al. (Int J Math Math Sci 24:371–377, 2000):
$$\begin{aligned} \left\{ \begin{array}{ll} \partial _tu+\alpha \partial _x^7u+\beta \partial _x^5u+\gamma \partial _x^3u+\mu \partial _xu+\lambda u\partial _xu=0,&{}\quad x \in {\mathbb {R}},\quad t\in {\mathbb {R}}, \\ u(x,0)=u_0(x),&{}\quad x\in {\mathbb {R}} \end{array}\right. \end{aligned}$$
with the initial data in the Sobolev space \(H^s({\mathbb {R}}).\) Benefited from the ideas of Huo and Jia (Z Angew Math Phys 59:634–646, 2008), Zhang et al. (Acta Math Sci 37B(2):385–394, 2017) and Zhang and Huang (Math Methods Appl Sci 39(10):2488–2513, 2016) that is, using Fourier restriction norm method, Tao’s [kZ]-multiplier method and the contraction mapping principle, we prove that IVP is locally well-posed for the initial data \(u_0\in H^s({\mathbb {R}})\) with \(s\ge -\frac{5}{8}\). Moreover, based on the local well-posedness and conservation law, we establish the global well-posedness for the initial data \(u_0\in H^s({\mathbb {R}})\) with \(s=0\).

Keywords

Higher order nonlinear dispersive equation Fourier restriction norm method Low regularity Well-posedness 

Mathematics Subject Classification

35A07 35Q53 

Notes

Acknowledgements

This work was supported by Hunan Provincial Natural Science Foundation of China Nos. 2016JJ2061, 2016JJ4037, Scientific Research Fund of Hunan Provincial Education Department Nos. 15B102, 15A077, NNSF of China Grant Nos. 11671101, 11371367, 11271118, the construct program of the key discipline in Hunan province No. 201176 and the Aid program for Science and Technology Innovative Research Team in Higher Educational Instituions of Hunan Province (No. 2014207)and the Hunan Provincial Local Cooperation Project of China Scholarship Council, Special Funds of Guangxi Distinguished Experts Construction Engineering.

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Authors and Affiliations

  1. 1.School of MathematicsHunan Institute of Science and TechnologyYueyangPeople’s Republic of China
  2. 2.College of SciencesGuangxi University for NationalitiesNanningPeople’s Republic of China
  3. 3.Guangxi Key Laboratory of Universities, Optimization Control and Engineering CalculationGuangxi University for NationalitiesGuangxiPeople’s Republic of China

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