{"paper":{"title":"Multi-dimensional Burgers equation with unbounded initial data: well-posedness and dispersive estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Denis Serre, Luis Silvestre","submitted_at":"2018-08-18T08:50:41Z","abstract_excerpt":"The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data $u_0$ is a bounded measurable function (Kruzhkov). The semi-group $(S_t)_{t\\ge0}$ is contracting in the $L^1$-distance. For the multi-dimensional Burgers equation, we show that $(S_t)_{t\\ge0}$ extends uniquely as a continuous semi-group over $L^p(\\mathbb{R}^n)$ whenever $1\\le p<\\infty$, and $u(t):=S_tu_0$ is actually an entropy solution to the Cauchy problem. When $p\\le q\\le \\infty$ and $t>0$, $S_t$ actually maps $L^p(\\mathbb{R}^n)$ into $L^q(\\mathbb{R}^n)$. These results are based upon new dispers"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.07467","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"}