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On the second-order asymptotic equation of a variational wave equation
Zhang, P; Zheng, YX
2002
Source PublicationPROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
ISSN0308-2105
Volume132Pages:483-509
AbstractWe 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.
Language英语
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000175394200014
PublisherROYAL SOC EDINBURGH
Citation statistics
Cited Times:11[WOS]   [WOS Record]     [Related Records in WOS]
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/17897
Collection中国科学院数学与系统科学研究院
Affiliation1.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
Recommended Citation
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.
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