{"paper":{"title":"Well-posedness and $L^1-L^p$ Smoothing Effect of the Porous Media Equation under Poincar\\'e Inequality","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Lukang Sun","submitted_at":"2025-04-08T06:45:24Z","abstract_excerpt":"We study the Cauchy problem for a weighted porous medium equation on $\\R$ associated with a Gibbs probability measure $\\pi=e^{-V}$. Under a Poincar\\'e inequality for $\\pi$ and the convexity assumption on $V$, we prove well-posedness and uniqueness of non-negative weak solutions with initial data in $L^1(\\R,\\pi)$. We also establish an $L^1$--$L^p$ smoothing effect at every positive time. More precisely, for every admissible $p>1$, we show that the logarithm of the ratio between the $L^p(\\R,\\pi)$ norm of the solution and its conserved $L^1(\\R,\\pi)$ mass first decays at a super-exponential rate a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.05722","kind":"arxiv","version":4},"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"}