{"paper":{"title":"Scalar conservation laws with rough (stochastic) fluxes; the spatially dependent case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Benoit Perthame (LJLL, INRIA-Paris-Rocquencourt), Panagiotis E. Souganidis, Pierre-Louis Lions (CEREMADE)","submitted_at":"2014-03-18T12:08:18Z","abstract_excerpt":"We continue the development of the theory of pathwise stochastic entropy solutions for scalar conservation laws in $\\R^N$ with quasilinear multiplicative ''rough path'' dependence by considering inhomogeneous fluxes and a single rough path like, for example, a Brownian motion. Following our previous note where we considered spatially independent fluxes, we introduce the notion of pathwise stochastic entropy solutions and prove that it is well posed, that is we establish existence, uniqueness and continuous dependence in the form of a (pathwise) $L^1$-contraction. Our approach is motivated by t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.4424","kind":"arxiv","version":2},"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"}