{"paper":{"title":"Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"\\'Angel Castro, Daniel Faraco, Francisco Mengual, Marcos Solera","submitted_at":"2025-02-14T16:32:20Z","abstract_excerpt":"We prove non-uniqueness of weak solutions to the forced $\\alpha$-SQG equation with Sobolev regularity $W^{s,p}$ in the supercritical regime $s < \\alpha + \\frac{2}{p}$, covering the 2D Euler equation ($\\alpha = 0$), the Surface Quasi-Geostrophic equation ($\\alpha = 1$), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show existence of global smooth solutions to the (unforced) $\\alpha$-SQG equation that are neither rotating nor traveling."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.10274","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/2502.10274/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"}