{"paper":{"title":"Semiconductor Boltzmann-Dirac-Benney equation with BGK-type collision operator: existence of solutions vs. ill-posedness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Marcel Braukhoff","submitted_at":"2017-11-16T10:44:34Z","abstract_excerpt":"A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice:\n  \\[\n  \\partial_t f + \\nabla_p\\epsilon(p)\\cdot\\nabla_x f - \\nabla_x n_f\\cdot\\nabla_p f = n_f(1- n_f)(\\mathcal{F}_f-f), \\quad x\\in\\mathbb{R}^d, p\\in\\mathbb{T}^d, t>0.\n  \\]\n  This system contains an interaction potential $n_f(x,t):=\\int_{\\mathbb{T}^d}f(x,p,t)dp$ being significantly more singular than the Coulomb potential, which is used in the Vlasov-Poisson system. This causes major structural difficulties in the analysis. Furthermore, $\\epsilon(p) ="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.06015","kind":"arxiv","version":3},"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"}