Asymptotic variational wave equations

doi:10.1007/s00205-006-0014-8

CSpace

	Asymptotic variational wave equations
	Bressan, Alberto ; Zhang, Ping ; Zheng, Yuxi
	2007
发表期刊	ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
ISSN	0003-9527
卷号	183 期号:1 页码:163-185
摘要	We investigate the equation (u(t) + (f(u))(x))(x) =f ''(u)(u(x) )(2)/2 where f(u) is a given smooth function. Typically f(u)=u(2)/2 or u(3)/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u(tt)-c(u)(c(u)u(x))(x)=0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view. We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.
DOI	10.1007/s00205-006-0014-8
语种	英语
WOS研究方向	Mathematics ; Mechanics
WOS类目	Mathematics, Interdisciplinary Applications ; Mechanics
WOS记录号	WOS:000241398900005
出版者	SPRINGER
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/4129
专题	中国科学院数学与系统科学研究院
通讯作者	Bressan, Alberto
作者单位	1.Penn State Univ, Dept Math, University Pk, PA 16802 USA 2.CAS, Acad Math & Syst Sci, Beijing 100080, Peoples R China
推荐引用方式 GB/T 7714	Bressan, Alberto,Zhang, Ping,Zheng, Yuxi. Asymptotic variational wave equations[J]. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS,2007,183(1):163-185.
APA	Bressan, Alberto,Zhang, Ping,&Zheng, Yuxi.(2007).Asymptotic variational wave equations.ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS,183(1),163-185.
MLA	Bressan, Alberto,et al."Asymptotic variational wave equations".ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 183.1(2007):163-185.