On the second-order asymptotic equation of a variational wave equation

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	On the second-order asymptotic equation of a variational wave equation
	Zhang, P ; Zheng, YX
	2002
发表期刊	PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
ISSN	0308-2105
卷号	132 页码:483-509
摘要	We have been interested in studying a nonlinear variational wave equation whose wave speed is a sinusoidal function of the wave amplitude, arising naturally from liquid crystals. High-frequency waves of small amplitudes, the so-called weakly nonlinear waves, near a constant state alpha are governed by two asymptotic equations: the first-order asymptotic equation if alpha is not a critical point of the sinusoidal function, or the second-order asymptotic equation if a is either a maximal or a minimal point of the sinusoidal function. Our earlier work on the first-order asymptotic equation has greatly helped the study of the nonlinear variational wave equation with monotone wave speed functions, It is apparent in our research that investigation of the second-order asymptotic equation is both crucial and equally illuminating for the study of the nonlinear variational wave equation with sinusoidal wave speed functions. We succeed in this paper in handling what may be appropriately called the 'concentration-annihilation' phenomena in the historical spirit of compensated-compactness (Tartar et al.), concentration-compactness (Lions), and concentration-cancellation or concentration-evanesces (DiPerna and Majda). More precisely, the second-order asymptotic equation has a product term uv(2) for which v(2) may have concentration on a set where u vanishes in a sequence of approximate solutions, while the product retains no concentration. Although absent in the first-order asymptotic equation, this concentration-annihilation phenomenon has been demonstrated through an explicit example for the nonlinear variational wave equation with sinusoidal wave speed functions in an earlier work. We use this concentration-annihilation to establish the global existence of weak solutions to the second-order asymptotic equation with initial data of bounded total variations.
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied ; Mathematics
WOS记录号	WOS:000175394200014
出版者	ROYAL SOC EDINBURGH
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/17897
专题	中国科学院数学与系统科学研究院
通讯作者	Zhang, P
作者单位	1.Acad Sinica, Inst Math, Beijing 100080, Peoples R China 2.Penn State Univ, Dept Math, University Pk, PA 16802 USA 3.Indiana Univ, Bloomington, IN 47405 USA
推荐引用方式 GB/T 7714	Zhang, P,Zheng, YX. On the second-order asymptotic equation of a variational wave equation[J]. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS,2002,132:483-509.
APA	Zhang, P,&Zheng, YX.(2002).On the second-order asymptotic equation of a variational wave equation.PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS,132,483-509.
MLA	Zhang, P,et al."On the second-order asymptotic equation of a variational wave equation".PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 132(2002):483-509.