On the model of the compressible hyperelastic rods and Euler equations on the circle

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:254

Article

Zhu, Min^2,3  Liu, Yue^1,4  Qu, Changzheng⁴ 

[1] Univ Texas Arlington, Dept Math, Arlington, TX 76019 USA

[2] Nanjing Forestry Univ, Dept Math, Nanjing 210037, Jiangsu, Peoples R China

[3] Southeast Univ, Dept Math, Nanjing 211189, Jiangsu, Peoples R China

[4] Ningbo Univ, Dept Math, Ningbo 315211, Zhejiang, Peoples R China

关键词: Hyperelastic rod equation;    Euler equation;    Camassa-Holm equation;    Diffeomorphisms group of the circle;   

DOI  :  10.1016/j.jde.2012.09.012

来源: Elsevier

【 摘 要 】

Considered herein is a geometric investigation on the one-parameter gamma-equations modeled in the cylindrical compressible hyperelastic rods. It is shown that the family of equations can only be realized as an Euler equation on the Lie group Diff(S-1) of all smooth and orientation preserving diffeomorphisms on the circle if the material parameter gamma = 1, which corresponds to the Camassa-Holm equation. In contrast, the Benjamin-Bona-Mahony (BBM) equation with the parameter gamma = 0 in this family of equations is not an Euler equation on Diff(S-1) for any inertia operator. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free