{"paper":{"title":"Lipschitz Metrics for a Class of Nonlinear Wave Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alberto Bressan, Geng Chen","submitted_at":"2015-06-21T01:48:24Z","abstract_excerpt":"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 w.r.t. the natural $H^1$ distance. 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 wei"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06310","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}