### Abstract

The nonlinear wave equation utt-c(u)(c(u)ux)x=0 determines a flow of conservative solutions taking values in the space H^{1}(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}(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 geodesic distance with the desired Lipschitz property.

Original language | English (US) |
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Pages (from-to) | 1303-1343 |

Number of pages | 41 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 226 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2017 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering

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## Cite this

*Archive for Rational Mechanics and Analysis*,

*226*(3), 1303-1343. https://doi.org/10.1007/s00205-017-1155-7