{"paper":{"title":"Global well-posedness and decay rates for the three dimensional incompressible active liquid crystals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Global strong solutions exist for the 3D active liquid crystal system when initial data are small and activity exceeds a critical threshold.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fan Yang, Xiongfeng Yang","submitted_at":"2026-05-06T08:13:36Z","abstract_excerpt":"This paper investigates the global well-posedness and large-time behavior of 3D incompressible active liquid crystals under constant activity, modeled by a coupled system of forced incompressible Navier-Stokes equations for the velocity and a parabolic system for the $Q$-tensor order parameter. By employing refined commutator estimates, the existence and uniqueness of global strong solutions are proved for small initial data $(Q_0,u_0)\\in H^{s+1}\\times H^s$ $(s\\geq 2)$ with activity $c>c_\\star$, which improves a previous result in \\cite{active-limit}. In addition, if the initial data further b"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Existence and uniqueness of global strong solutions for small initial data (Q0,u0) in H^{s+1} x H^s (s>=2) when activity c > c_star, together with a mixing decay estimate on partial^k Q(t) that combines exponential decay at rate proportional to (c - c_star) Gamma and the optimal algebraic heat-kernel rate for k <= s-1.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The initial data must be sufficiently small in the indicated Sobolev norms and the activity must exceed the critical threshold c_star; the proof relies on this smallness to close the a priori estimates via refined commutator bounds.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Global well-posedness for small data and activity-dependent decay rates are established for the 3D Beris-Edwards active liquid crystal system using commutator estimates and Green's functions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Global strong solutions exist for the 3D active liquid crystal system when initial data are small and activity exceeds a critical threshold.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b63712d7f4041f7545fd95b22d3e4427deb12acfb081b7a2566c29c93fc45ede"},"source":{"id":"2605.04625","kind":"arxiv","version":2},"verdict":{"id":"125550d8-6c9c-4876-ab93-c8be90d42ff6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:51:43.849802Z","strongest_claim":"Existence and uniqueness of global strong solutions for small initial data (Q0,u0) in H^{s+1} x H^s (s>=2) when activity c > c_star, together with a mixing decay estimate on partial^k Q(t) that combines exponential decay at rate proportional to (c - c_star) Gamma and the optimal algebraic heat-kernel rate for k <= s-1.","one_line_summary":"Global well-posedness for small data and activity-dependent decay rates are established for the 3D Beris-Edwards active liquid crystal system using commutator estimates and Green's functions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The initial data must be sufficiently small in the indicated Sobolev norms and the activity must exceed the critical threshold c_star; the proof relies on this smallness to close the a priori estimates via refined commutator bounds.","pith_extraction_headline":"Global strong solutions exist for the 3D active liquid crystal system when initial data are small and activity exceeds a critical threshold."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04625/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:31:19.957041Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:18:02.735892Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"bbbc4a59ab60a6c4ad458d4ec46453250e84785ca431e4bfa92bc3928bb5d506"},"references":{"count":69,"sample":[{"doi":"","year":2014,"title":"H. Abels, G. Dolzmann and Y . N. Liu,Well-posedness of a fully coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data, SIAM Journal on Mathematical Analysis, 2014, 46(4): 3050–3077","work_id":"5cad5379-a61b-4aff-aeb0-2c140f039844","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"H. Abels, G. Dolzmann and Y . N. Liu,Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions, Advances in Differential Equations, 2016, ","work_id":"96390446-b1a5-4aab-8513-8aa58b4c05e0","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"De Anna,A global 2D well-posedness result on the order tensor liquid crystal theory, Journal of Differential Equations, 2017, 262(7): 3932–3979","work_id":"91bee664-b4af-4dfd-9771-817f655d6c46","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"De Anna and A","work_id":"d8ddf72a-8f24-42fb-bd08-0ef9cf6b2aaa","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"J. M. Ball and A. Majumdar,Nematic liquid crystals: from Maier-Saupe to a continuum theory, Molecular crystals and liquid crystals, 2010, 525(1): 1–11","work_id":"7b8ac1d5-9bd3-4739-b02c-6d56cd9cd9d5","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":69,"snapshot_sha256":"b604bf880723c93b547820e848ad66e1237a798fe0e0811d5157a0abe1ab92fe","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"ce03c391e41f1a6c3e8fb34754ef3b28ce5d725b9ace10e5db1abeaed49afd99"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}