Lipschitz Metrics for a Class of Nonlinear Wave Equations
作者: Alberto BressanGeng Chen
作者单位: 1Penn State University
2University of Kansas
刊名: Archive for Rational Mechanics and Analysis, 2017, Vol.226 (3), pp.1303-1343
来源数据库: Springer Nature Journal
DOI: 10.1007/s00205-017-1155-7
原始语种摘要: The nonlinear wave equation \({u_{tt}-c(u)(c(u)u_x)_x=0}\) determines a flow of conservative solutions taking values in the space \({H^1(\mathbb{R})}\) . However, this flow is not continuous with respect to the natural H 1 distance. The aim of this paper is to construct a new metric which renders the flow uniformly Lipschitz continuous on bounded subsets of \({H^1(\mathbb{R})}\) . For this purpose, H 1 is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, one can carefully estimate how the weighted length grows in time. By the generic regularity result proved in [ 7 ], these piecewise regular paths are dense and can be used to construct a...
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影响因子:2.292 (2012)

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关键词翻译
关键词翻译
  • geodesic 测地的
  • piecewise 分段
  • construct 建设
  • regularity 规则性
  • tangent 切线
  • bounded 有界的
  • weighted 加权
  • continuous 连续的
  • respect 珍视
  • uniformly 一律地