KMS Of Academy of mathematics and systems sciences, CAS
On the second-order asymptotic equation of a variational wave equation | |
Zhang, P; Zheng, YX | |
2002 | |
Source Publication | PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS |
ISSN | 0308-2105 |
Volume | 132Pages:483-509 |
Abstract | 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. |
Language | 英语 |
WOS Research Area | Mathematics |
WOS Subject | Mathematics, Applied ; Mathematics |
WOS ID | WOS:000175394200014 |
Publisher | ROYAL SOC EDINBURGH |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/17897 |
Collection | 中国科学院数学与系统科学研究院 |
Affiliation | 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 |
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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