{"paper":{"title":"From Polynomial Stability to Periodic Well-posedness in Partially Dissipative Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Polynomial stability of the semigroup yields an explicit characterization of the dense forcing set on which periodic well-posedness holds via resolvent bounds that dictate required losses of time derivatives on the forcing.","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Boris Muha, Giovanni P. Galdi, Justin T. Webster","submitted_at":"2026-05-13T02:15:15Z","abstract_excerpt":"The study of resonances (and well-posedness) for complex systems under time-periodic loading is of broad interest in application. The work of Galdi et al.~(2014) connects asymptotic stability of solutions to an unforced Cauchy problem to solvability of the time-periodic forced problem. Uniform stability of the solution semigroup gives periodic well-posedness for all forces in the natural mild forcing class, whereas strong stability yields only existence of a dense set of forcings for which resonance can be excluded. We address an intermediate regime for polynomial (also: rational or semiunifor"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Polynomial stability of the semigroup yields an explicit characterization of the dense forcing set on which periodic well-posedness holds; resolvent bounds translate directly into certain losses of time derivatives on the forcing required to ensure well-posedness.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The systems admit a Fourier decomposition in Hilbert space that converts polynomial decay rates into explicit resolvent bounds controlling the forcing class.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Polynomial semigroup stability implies periodic well-posedness on a dense set of forcings whose time-derivative loss is controlled by resolvent bounds.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Polynomial stability of the semigroup yields an explicit characterization of the dense forcing set on which periodic well-posedness holds via resolvent bounds that dictate required losses of time derivatives on the forcing.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5e5d9b979edba33307745bbd28756424da3844c3fdc4d36d6caac49e2805818d"},"source":{"id":"2605.12892","kind":"arxiv","version":1},"verdict":{"id":"f4e54668-49aa-4d9b-9ed9-b5190bee35c3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:56:04.138560Z","strongest_claim":"Polynomial stability of the semigroup yields an explicit characterization of the dense forcing set on which periodic well-posedness holds; resolvent bounds translate directly into certain losses of time derivatives on the forcing required to ensure well-posedness.","one_line_summary":"Polynomial semigroup stability implies periodic well-posedness on a dense set of forcings whose time-derivative loss is controlled by resolvent bounds.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The systems admit a Fourier decomposition in Hilbert space that converts polynomial decay rates into explicit resolvent bounds controlling the forcing class.","pith_extraction_headline":"Polynomial stability of the semigroup yields an explicit characterization of the dense forcing set on which periodic well-posedness holds via resolvent bounds that dictate required losses of time derivatives on the forcing."},"references":{"count":22,"sample":[{"doi":"","year":2014,"title":"G. P. Galdi, M. Mohebbi, R. Zakerzadeh, P. Zunino, Hyperbolic–parabolic coupling and the occurrence of resonance in partially dissipative systems, in: Fluid-Structure Interaction and Biomedical Applic","work_id":"869f7046-f05c-4aae-a944-99df0c76ea31","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1973,"title":"I. Straˇ skraba, O. Vejvoda, Periodic solutions to abstract differential equations, Czechoslovak Math. J. 23 (1973) 635–669","work_id":"ffaccc1d-f983-41c9-9082-11126f555e1f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1985,"title":"Haraux, Non-resonance for a strongly dissipative wave equation in higher dimensions, Manuscripta Math","work_id":"a4afd82c-3afe-4eaf-97a6-83cbbdd3c8a5","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"A. Borichev, Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347 (2010) 455–478","work_id":"23b3cfb3-8e69-4e75-a936-53603b31c4e2","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1984,"title":"Pr¨ uss, On the spectrum ofC 0-semigroups, Trans","work_id":"619bd88f-c0fc-4c48-a136-a1fbedd97866","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":22,"snapshot_sha256":"30a19d54742534766fd99fd8315cbc753bdbb49f4cf20cf6a58eebc293d8c1d3","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"}