{"paper":{"title":"The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Martin Kohlmann","submitted_at":"2011-07-20T15:09:12Z","abstract_excerpt":"In this paper, the two-dimensional periodic $b$-equation is discussed under geometric aspects, i.e., as a geodesic flow on the diffeomorphism group of the torus $\\T=S^1\\times S^1$. In the framework of Arnold's [V.I. Arnold, Sur la g\\'eom\\'etrie diff\\'erentielle des groupes de Lie de dimension infinie et ses applications \\`a l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16 (1966) 319-361] famous approach, we achieve some well-posedness results for the $b$-equation and we perform explicit curvature computations for the 2D Camassa-Holm equation, which is obtained for $b=2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3998","kind":"arxiv","version":2},"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"}