{"paper":{"title":"Small scale formation for the 2D Boussinesq equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexander Kiselev, Jaemin Park, Yao Yao","submitted_at":"2022-11-09T18:00:46Z","abstract_excerpt":"We study the 2D incompressible Boussinesq equation without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global smooth solutions satisfying $\\sup_{\\tau\\in[0,t]} \\|\\nabla \\rho(\\tau)\\|_{L^2}\\gtrsim t^\\alpha$ for some $\\alpha>0$. For the inviscid equation in the strip, we construct examples satisfying $\\|\\omega(t)\\|_{L^\\infty}\\gtrsim t^3$ and $\\sup_{\\tau\\in[0,t]} \\|\\nabla \\rho(\\tau)\\|_{L^\\infty} \\gtrsim t^2$ during the existence of a smooth solution. These growth results hold for a broad "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.05070","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.05070/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}