{"paper":{"title":"On well/ill-posedness for the generalized surface quasi-geostrophic equations in H\\\"older spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jinwook Jung, Junha Kim, Young-Pil Choi","submitted_at":"2024-05-02T12:49:47Z","abstract_excerpt":"We establish the well/ill-posedness theories for the inviscid $\\alpha$-surface quasi-geostrophic ($\\alpha$-SQG) equations in H\\\"older spaces, where $\\alpha = 0$ and $\\alpha = 1$ correspond to the two-dimensional Euler equation in the vorticity formulation and SQG equation of geophysical significance, respectively. We first prove the local-in-time well-posedness of $\\alpha$-SQG equations in $C([0,T);C^{0,\\beta}(\\mathbb{R}^2))$ with $\\beta \\in (\\alpha,1)$ for some $T>0$. We then analyze the strong ill-posedness in $C^{0,\\alpha}(\\mathbb{R}^2)$ constructing smooth solutions to the $\\alpha$-SQG equ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.01245","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/2405.01245/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"}