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

• DOI: 10.1007/s10884-018-9669-8
• Publication Date: 2018-05-03T08:33:04Z
• Authors: Zaiyun Zhang; Zhenhai Liu; Mingbao Sun; Songhua Li
• Keywords: Conservation law

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 [k, Z]-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 $$s=0$$ .