Weak solutions to a nonlinear variational wave equation with general data

doi:10.1016/j.anihpc.2004.04.001

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	Weak solutions to a nonlinear variational wave equation with general data
	Zhang, P ; Zheng, YX
	2005
发表期刊	ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
ISSN	0294-1449
卷号	22 期号:2 页码:207-226
摘要	We establish the existence of global weak solutions to the initial value problem for a nonlinear variational wave equation u(tt) - c(u)(c(u)u(x))(x) = 0 with general initial data (u(0), u(t) (0)) = (u(0), u(1)) is an element of W-1.2 x L-2 under the assumptions that the wave speed c(u) satisfies c '((.)) >= 0 and c '(u(0)((.))) > 0. Moreover, we obtain high regularity for the spatial derivative partial derivative(x)u of the wave amplitude u away from where c(u) = 0. This equation arises from studies in nematic liquid crystals, long waves on a dipole chain, and a few other fields. We use Young measure method in the setting of LP spaces and method of renormalization to overcome the difficulty that oscillations in a sequence of approximations get amplified by the quadratic growth term of the equation. We use a high space-time estimate for partial derivative(x)u to handle possible concentrations. This result improves our earlier existence result for initial data in the space W-1,W-infinity x L-infinity to the natural space W-1,W-2 x L-2. (c) 2004 Elsevier SAS. All rights reserved.
DOI	10.1016/j.anihpc.2004.04.001
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000227722200005
出版者	GAUTHIER-VILLARS/EDITIONS ELSEVIER
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/2376
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.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100080, Peoples R China 2.Penn State Univ, Dept Math, University Pk, PA 16802 USA
推荐引用方式 GB/T 7714	Zhang, P,Zheng, YX. Weak solutions to a nonlinear variational wave equation with general data[J]. ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE,2005,22(2):207-226.
APA	Zhang, P,&Zheng, YX.(2005).Weak solutions to a nonlinear variational wave equation with general data.ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE,22(2),207-226.
MLA	Zhang, P,et al."Weak solutions to a nonlinear variational wave equation with general data".ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE 22.2(2005):207-226.