Lipschitz Metrics for a Class of Nonlinear Wave Equations

Alberto Bressan, Geng Chen

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


The nonlinear wave equation utt-c(u)(c(u)ux)x=0 determines a flow of conservative solutions taking values in the space H1(R). However, this flow is not continuous with respect to the natural H1 distance. The aim of this paper is to construct a new metric which renders the flow uniformly Lipschitz continuous on bounded subsets of H1(R). For this purpose, H1 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 geodesic distance with the desired Lipschitz property.

Original languageEnglish (US)
Pages (from-to)1303-1343
Number of pages41
JournalArchive for Rational Mechanics and Analysis
Issue number3
StatePublished - Dec 1 2017

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering


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