{"paper":{"title":"On the monotonicity of the entropy production in the Landau-Maxwell equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The entropy production of the homogeneous Landau-Maxwell equation becomes non-increasing after a finite time under well-distributed directional temperatures and sufficient moments.","cross_cats":[],"primary_cat":"math.AP","authors_text":"C\\^ome Tabary","submitted_at":"2026-01-06T15:37:27Z","abstract_excerpt":"We study the homogeneous Landau equation with Maxwell molecules and prove that the entropy production is non-increasing provided the directional temperatures are well-distributed and the solution admits a moment of order $\\ell$, for some $\\ell$ arbitrarily close to $2$. It implies that for an initial condition with finite moment of order $\\ell$, the entropy production is guaranteed to be non-increasing after a certain time, that we explicitly compute. This is the first partial answer to a conjecture made by Henry P. McKean in 1966 on the sign of the time-derivatives of the entropy. Without mom"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We study the homogeneous Landau equation with Maxwell molecules and prove that the entropy production is non-increasing provided the directional temperatures are well-distributed and the solution admits a moment of order ℓ, for some ℓ arbitrarily close to 2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The directional temperatures are well-distributed and the solution admits a moment of order ℓ arbitrarily close to 2; if this moment condition fails the monotonicity may not hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Entropy production for the Landau equation with Maxwell molecules is non-increasing after a finite time under moment and temperature-distribution conditions, partially resolving a 1966 conjecture.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The entropy production of the homogeneous Landau-Maxwell equation becomes non-increasing after a finite time under well-distributed directional temperatures and sufficient moments.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b5b21b1f113a2d3cfa03fe82a359bd3d5bc3836411bb8e896f32d34e1cf577f8"},"source":{"id":"2601.03107","kind":"arxiv","version":3},"verdict":{"id":"b1b2e340-76f8-448a-a2a4-85cf0887a611","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T17:07:06.813983Z","strongest_claim":"We study the homogeneous Landau equation with Maxwell molecules and prove that the entropy production is non-increasing provided the directional temperatures are well-distributed and the solution admits a moment of order ℓ, for some ℓ arbitrarily close to 2.","one_line_summary":"Entropy production for the Landau equation with Maxwell molecules is non-increasing after a finite time under moment and temperature-distribution conditions, partially resolving a 1966 conjecture.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The directional temperatures are well-distributed and the solution admits a moment of order ℓ arbitrarily close to 2; if this moment condition fails the monotonicity may not hold.","pith_extraction_headline":"The entropy production of the homogeneous Landau-Maxwell equation becomes non-increasing after a finite time under well-distributed directional temperatures and sufficient moments."},"references":{"count":31,"sample":[{"doi":"","year":1975,"title":"Fourier Transform Method in the Theory of the Boltzmann Equa- tion for Maxwellian Molecules","work_id":"3be5bec8-c1d4-4941-a121-d5c391469692","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.48550/arxiv.2504.13802","year":2025,"title":"F.-U. Caja-Lopez, M. G. Delgadino, M.-P . Gualdani, and M. Taskovic.Contractivity of Wasserstein Distance and Exponential Decay for the Landau Equation with Maxwellian Molecules. Nov. 2025.DOI:10.4855","work_id":"342b82ac-2b47-48c9-b564-afcacc61e278","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.3934/krm.2016.9.1(cit","year":2015,"title":"Propagation of Chaos for the Spatially Homogeneous Landau Equation for Maxwellian Molecules","work_id":"bced18c0-37f8-462b-8518-3b23ff8b2240","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"José Antonio Carrillo and Shuchen Guo.From Fisher Information Decay for the Kac Model to the Landau-Coulomb Hierarchy. Feb. 2025.DOI:10 . 48550 / arXiv . 2502 . 18606. arXiv:2502.18606 [math](cit. on ","work_id":"c4d14ad2-8b4c-4c1f-b448-fbade81d4d03","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"On the Spatially Homogeneous Landau Equation for Hard Potentials Part Ii : H-Theorem and Applications","work_id":"c5c0f29c-443a-4a51-8be7-058618429cde","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":31,"snapshot_sha256":"5e4423d0200a8e7716e5456d15d695d57f57da93ad218a6112f0638d7a503282","internal_anchors":3},"formal_canon":{"evidence_count":1,"snapshot_sha256":"e196cdfa4722fe8ea32b097dee0040eea4ae2e7903421bd754c3e413efc553a2"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}