{"paper":{"title":"On the Schr\\\"odinger-Debye System in Compact Riemannian Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mahendra Panthee, Marcelo Nogueira","submitted_at":"2018-10-30T14:55:08Z","abstract_excerpt":"We consider the initial value problem (IVP) associated to the Schr\\\"odinger-Debye system posed on a $d$-dimensional compact Riemannian manifold $M$ and prove local well-posedness result for given data $(u_0, v_0)\\in H^s(M)\\times (H^s(M)\\cap L^{\\infty}(M))$ whenever $s>\\frac{d}2-\\frac12$, $d\\geq 2$. For $d=2$, we apply a sharp version of the Gagliardo-Nirenberg inequality in compact manifold to derive an a priori estimate for the $H^1$-solution and use it to prove the global well-posedness result in this space."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.12788","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"}