{"paper":{"title":"Multiple Vortices for the Shallow Water Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daomin Cao, Zhongyuan Liu","submitted_at":"2013-01-28T00:57:19Z","abstract_excerpt":"In this paper, we construct stationary classical solutions of the shallow water equation with vanishing Froude number $Fr$ in the so-called lake model.\n  To this end we need to study solutions to the following semilinear elliptic problem \\[{cases} -\\varepsilon^2\\text{div}(\\frac{\\nabla u}{b})=b(u-q\\log\\frac{1}{\\varepsilon})_+^{p},& \\text{in}\\; \\Omega, u=0, &\\text{on}\\;\\partial \\Omega, {cases} \\] for small $\\varepsilon>0$, where $p>1$, $\\text{div}(\\frac{\\nabla q}{b})=0$ and $\\Omega\\subset\\mathbb{R}^2$ is a smooth bounded domain,.\n  We showed that if $\\frac{q^2}{b}$ has $m$ strictly local minimum"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6420","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"}