{"paper":{"title":"On the one-dimensional continuity equation with a nearly incompressible vector field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nikolay A. Gusev","submitted_at":"2016-10-27T15:46:16Z","abstract_excerpt":"We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field $b\\colon (0,T) \\times \\mathbb R^d \\to \\mathbb R^d$, $T>0$. This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system).\n  It is well known that in the generic multi-dimensional case ($d\\ge 1$) near incompressibility is sufficient for existence of bounded weak solutions, but uniqueness may fail (even when the vector field is divergence-free), and hence further assumptions on the regularity of $b$ (e.g. Sobolev regularity) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.08848","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"}