{"paper":{"title":"Singularity Formation and Global Well-Posedness for the Generalized Constantin-Lax-Majda Equation with Dissipation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiajie Chen","submitted_at":"2019-08-25T20:06:07Z","abstract_excerpt":"We study a generalization due to De Gregorio and Wunsch et.al. of the Constantin-Lax-Majda equation (gCLM) on the real line \\[ \\omega_t + a u \\omega_x = u_x \\omega - \\nu \\Lambda^{\\gamma} \\omega, \\quad u_x = H \\omega , \\] where $H$ is the Hilbert transform and $\\Lambda = (-\\partial_{xx})^{1/2}$. We use the method in \\cite{chen2019finite} to prove finite time self-similar blowup for $a$ close to $\\frac{1}{2}$ and $\\gamma=2$ by establishing nonlinear stability of an approximate self-similar profile. For $a>-1$, we discuss several classes of initial data and establish global well-posedness and an "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09385","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/1908.09385/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"}