Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation
Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation
复制标题
DOI：
10.1007/s00205-011-0403-5
复制
发表时间：
2009-10
影响因子：
2.5
通讯作者：
H. Holden;X. Raynaud
中科院分区：
文献类型：
--
作者：
H. Holden;X. Raynaud
证明了非线性变分波动方程utt −c（u）（c（u）ux）x= 0的保守解的整体半群的存在性.我们考虑到初始数据，|t= 0andut| t= 0，包含度量。我们假设。这个方程的解可能会经历能量密度集中到零测度的集合中。通过引入与特征相关的新变量来构造解决方案，从而使能量密度中的奇异性变得易于管理。此外，我们还证明了能量只可能集中在零测度的一组时间上或集中在c ′（u）为零的点上.给出了一种新的构造守恒解的数值方法，并举例说明。
We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equationutt−c(u)(c(u)ux)x= 0. We allow for initial datau|t= 0andut|t=0that contain measures. We assume that. Solutions of this equation may experience concentration of the energy densityinto sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points wherec′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.