NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

doi:10.1016/S0252-9602(14)60123-5

数学物理学报(英文版) ›› 2014, Vol. 34 ›› Issue (6): 1781-1794.doi: 10.1016/S0252-9602(14)60123-5

NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

傅仰耿|刘正荣|唐昊

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China； Department of Mathematics, South China University of Technology, Guangzhou 510640, China

收稿日期:2013-08-21 修回日期:2014-01-28 出版日期:2014-11-20 发布日期:2014-11-20
基金资助:
This work is supported by the National Natural Science Foundation of China (11226159).

NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

FU Yang-Geng, LIU Zheng-Rong, TANG Hao

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China； Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received:2013-08-21 Revised:2014-01-28 Online:2014-11-20 Published:2014-11-20
Supported by:
This work is supported by the National Natural Science Foundation of China (11226159).

摘要/Abstract

摘要：

In this paper, we study the Cauchy problem for the modified Camassa-Holm equation
mt + um_x+ 2u_xm = 0, m = (1 − ∂²_x)²u,
u(x, 0) = u₀(x) ∈ H^s(R), x ∈R, t > 0,
and show that the solution map is not uniformly continuous in Sobolev spaces Hs(R) for s > 7/2. Compared with the periodic problem, the non-periodic problem is more difficult, e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.

关键词: modified Camassa-Holm equation, Cauchy problem, non-uniform continuity, Sobolev spaces

Abstract:

Key words: modified Camassa-Holm equation, Cauchy problem, non-uniform continuity, Sobolev spaces

中图分类号:

35A07

傅仰耿, 刘正荣, 唐昊. NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE[J]. 数学物理学报(英文版), 2014, 34(6): 1781-1794.

FU Yang-Geng, LIU Zheng-Rong, TANG Hao. NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE[J]. Acta mathematica scientia,Series B, 2014, 34(6): 1781-1794.

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NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

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