{"paper":{"title":"Logarithmic Sobolev inequalities and spectral concentration for the cubic Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.SP","authors_text":"(2) University of New South Wales), Caroline Brett (1), Gordon Blower (1), Ian Doust (2) ((1) Lancaster University","submitted_at":"2013-08-16T15:14:30Z","abstract_excerpt":"The nonlinear Schr\\\"odinger equation NLSE(p, \\beta), -iu_t=-u_{xx}+\\beta | u|^{p-2} u=0, arises from a Hamiltonian on infinite-dimensional phase space \\Lp^2(\\mT). For p\\leq 6, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that there exists a Gibbs measure \\mu^{\\beta}_N on balls \\Omega_N= {\\phi \\in \\Lp^2(\\mT) : | \\phi |^2_{\\Lp^2} \\leq N} in phase space such that the Cauchy problem for NLSE(p,\\beta) is well posed on the support of \\mu^{\\beta}_N, and that \\mu^{\\beta}_N is invariant under the flow. This paper shows that \\mu^{\\beta}_N satisfies a logarithmic Sobolev inequality for the fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.3649","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"}