{"paper":{"title":"Local regularity for the modified SQG patch equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexander Kiselev, Andrej Zlatos, Yao Yao","submitted_at":"2015-08-30T18:18:21Z","abstract_excerpt":"We study the patch dynamics on the whole plane and on the half-plane for a family of active scalars called modified SQG equations. These involve a parameter $\\alpha$ which appears in the power of the kernel in their Biot-Savart laws and describes the degree of regularity of the equation. The values $\\alpha=0$ and $\\alpha=\\frac 12$ correspond to the 2D Euler and SQG equations, respectively. We establish here local-in-time regularity for these models, for all $\\alpha\\in(0,\\frac 12)$ on the whole plane and for all small $\\alpha>0$ on the half-plane. We use the latter result in [16], where we show"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07611","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"}