JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (8): 42-49.doi: 10.6040/j.issn.1671-9352.0.2019.125

Previous Articles     Next Articles

Non-uniform dependence on initial data for the periodic two-coupled Camassa-Holm system

WANG Hai-quan   

  1. School of Mathematics, Northwest University, Xian 710127, Shaanxi, China
  • Online:2019-08-20 Published:2019-07-03

PDF (PC)

293

Abstract: Considered herein is the initial value problem for the periodic two-coupled Camassa-Holm system. It is shown that the solution map of this problem is not uniformly continuous in Besov spaces B3/22,1(T)×B3/22,1(T). Based on the well-posedness result and the lifespan for this problem, the method of approximate solutions is utilized. The same approach can be used to discuss this property of the solutions for the other nonlinear partial differential equations.

Key words: the periodic two-coupled Camassa-Holm system, non-uniform dependence, Besov spaces, approximate solutions

CLC Number: 

  • O175.2
[1] FU Ying, QU Changzheng. Well-posedness and blow-up solution for a new coupled Camassa-Holm equations[J]. Journal of Mathematical Physics, 2009, 50(1): 012906.
[2] ZHU Mingxuan. Blow-up, global existence and persistence properties for the coupled Camassa-Holm equation[J]. Mathematical Physics Analysis and Geomery, 2011, 14(3): 197-209.
[3] TIAN Lixin, YAN Wanfeng, GUI Guilong. On the local well-posedness and blow-up solution to coupled Camassa-Holm equations in Besov spaces[J]. Journal of Mathematical Physics, 2012, 53(1): 013701.
[4] LIU Xingxing. Local well-posedness of a coupled Camassa-Holm system in critical spaces[J]. Zeitschrift für Analysis und ihre Anwendungen, 2015, 34(1): 43-59.
[5] HIMONAS A, KENIG C, MISIOŁEK G. Non-uniform dependence for the periodic CH equation[J]. Communications in Partial Differential Equations, 2010, 35(6): 1145-1162.
[6] HIMONAS A, MISIOŁEK G. High-frequency smooth solutions and well-posedness of the Camassa-Holm equation[J]. Interbational Mathematics Research Notices, 2005, 2005(51): 3135-3151.
[7] TANG Hao, ZHAO Yongye, LIU Zhengrong. A note on the solution map for the periodic Camassa-Holm equation[J]. Applicable Analysis, 2014, 93(8): 1745-1760.
[8] BAHOURI H, CHEMIN J, DANCHIN R. Fourier analysis and nonlinear partial differential equations[M]. Berlin: Springer-Verlag, 2011.
[9] DANCHIN R. A few remarks on the Camassa-Holm equation[J]. Differentialand Integral Equations, 2001, 14(8): 953-988.
[10] LI Jinlu, YIN Zhaoyang. Well-posedness and analytic solutions of two-component Euler-Poincaré system[J]. Monatshefte für Mathematik, 2017, 183(3): 509-537.
[11] DANCHIN R. A note on well-posedness for Camassa-Holm equation[J]. Journal of Differential Equations, 2003, 192(2): 429-444.
[12] DANCHIN R. Fourier analysis methods for PDEs[R/OL].(2005-11-14)[2019-01-20]. https://m.doc88.com/p-9079718125760.html.
[1] CAO Kai-kai. Wavelet estimation for anisotropic density functions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 99-105.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] TANG Feng-qin1, BAI Jian-ming2. The precise large deviations for a risk model with extended negatively upper orthant dependent claim  sizes[J]. J4, 2013, 48(1): 100 -106 .
[2] QIU Tao-rong, WANG Lu, XIONG Shu-jie, BAI Xiao-ming. A granular computing approach for knowledge hiding[J]. J4, 2010, 45(7): 60 -64 .
[3] XUE Qiu-fang1,2, GAO Xing-bao1*, LIU Xiao-guang1. Several equivalent conditions for H-matrix based on the extrapolated GaussSeidel iterative method[J]. J4, 2013, 48(4): 65 -71 .
[4] SUN Xiao-ting1, JIN Lan2*. Application of DOSY in oligosaccharide mixture analysis[J]. J4, 2013, 48(1): 43 -45 .
[5] REN Min1,2, ZHANG Guang-hui1. Absorbing probabilities of random walks in an independent random  environment convergence in distribution on the half-line[J]. J4, 2013, 48(1): 93 -99 .
[6] YU Xiu-qing. (σ,τ)-expansion model of P-sets and its properties#br#[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(04): 90 -94 .
[7] WANG Qi,ZHAO Xiu-heng,LI Guo-jun . Embedding hypergraph in trees of rings[J]. J4, 2007, 42(10): 114 -117 .
[8] QANG Yao,LIU Jian and WANG Ren-qing,* . Allee effect and its significance to small population management in nature reservation and biological invasions[J]. J4, 2007, 42(1): 76 -82 .
[9] ZHAO Jun1, ZHAO Jing2, FAN Ting-jun1*, YUAN Wen-peng1,3, ZHANG Zheng1, CONG Ri-shan1. Purification and anti-tumor activity examination of water-soluble asterosaponin from Asterias rollestoni Bell[J]. J4, 2013, 48(1): 30 -35 .
[10] YANG Yong-wei1, 2, HE Peng-fei2, LI Yi-jun2,3. On strict filters of BL-algebras#br#[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 63 -67 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd