{"paper":{"title":"A short proof of the large time energy growth for the Boussinesq system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Charafeddine Mouzouni, Lorenzo Brandolese","submitted_at":"2017-03-02T14:11:35Z","abstract_excerpt":"We give a direct proof of the fact that the $L^{p)$-norms of global solutions of the Boussinesq system in $R^{3}$ grow large as $ t \\rightarrow + \\infty $ for $ 1 < p < 3 $ and decay to zero for $ 3 < p \\leq \\infty $, providing exact estimates from below and above using a suitable decomposition of the space-time space $ R^{+} \\times R^{3} $. In particular, the kinetic energy blows up as $ \\| u(t) \\|_{2}^{2} \\sim c t^{1/2} $ for large time. This contrasts with the case of the Navier-Stokes equations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00793","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"}