{"paper":{"title":"On existence of local and global strong solutions for the stochastic tamed Navier-Stokes equations on $\\mathbb{R}^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Stochastic tamed Navier-Stokes equations on R^3 admit pathwise unique maximal local strong solutions for initial data in L^p with p greater than 3, and unique global solutions with added H^1 regularity.","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Bikram Podder, Surendra Kumar","submitted_at":"2026-05-05T13:22:29Z","abstract_excerpt":"We study the existence of local and global strong solutions for the stochastic tamed Navier--Stokes equations on the whole space $\\mathbb{R}^3$, driven by multiplicative Wiener noise and compensated L\\'evy jump noise. For $p > 3$, we first prove the existence of a pathwise unique maximal local $L^p$-strong solution for divergence-free, $\\mathcal{F}_0$-measurable initial data in $L^p(\\Omega; L^p(\\mathbb{R}^3;\\mathbb{R}^3))$. For initial data additionally belonging to $L^2(\\Omega; H^1(\\mathbb{R}^3;\\mathbb{R}^3))$, we overcome the non-local pressure obstruction inherent to the whole space, to est"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we first prove the existence of a pathwise unique maximal local strong solution for F_0-measurable initial data in L^p(Ω; L^p(R^3; R^3)) for p > 3. Furthermore, by assuming initial data in L^p(Ω; L^p(R^3; R^3)) ∩ L^2(Ω; H^1(R^3; R^3)), we overcome the non-local pressure obstruction to establish the existence of a unique global strong solution.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The initial data must satisfy the stated integrability conditions in L^p for p>3 (and additional H^1 regularity for the global case); the taming mechanism is assumed to be sufficient to control the nonlinearity and allow continuation past potential singularities.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Pathwise unique maximal local strong solutions exist for initial data in L^p(Ω; L^p(R^3)) with p>3, and unique global strong solutions exist when initial data also lies in L^2(Ω; H^1(R^3)).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Stochastic tamed Navier-Stokes equations on R^3 admit pathwise unique maximal local strong solutions for initial data in L^p with p greater than 3, and unique global solutions with added H^1 regularity.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7f13cd471e3c907cad25a096e38551d05e94db98c2615e914f4fa038f9e88c5a"},"source":{"id":"2605.03734","kind":"arxiv","version":2},"verdict":{"id":"e1b82885-0b4a-4a3a-a434-988b45a3be64","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T14:40:15.795545Z","strongest_claim":"we first prove the existence of a pathwise unique maximal local strong solution for F_0-measurable initial data in L^p(Ω; L^p(R^3; R^3)) for p > 3. Furthermore, by assuming initial data in L^p(Ω; L^p(R^3; R^3)) ∩ L^2(Ω; H^1(R^3; R^3)), we overcome the non-local pressure obstruction to establish the existence of a unique global strong solution.","one_line_summary":"Pathwise unique maximal local strong solutions exist for initial data in L^p(Ω; L^p(R^3)) with p>3, and unique global strong solutions exist when initial data also lies in L^2(Ω; H^1(R^3)).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The initial data must satisfy the stated integrability conditions in L^p for p>3 (and additional H^1 regularity for the global case); the taming mechanism is assumed to be sufficient to control the nonlinearity and allow continuation past potential singularities.","pith_extraction_headline":"Stochastic tamed Navier-Stokes equations on R^3 admit pathwise unique maximal local strong solutions for initial data in L^p with p greater than 3, and unique global solutions with added H^1 regularity."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.03734/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T13:35:12.325291Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T00:31:21.290744Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T15:03:56.657271Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"be9365c94a7c5efc8ddcc328a2d2a682f3a4f6c26258d8c45c556929e95351d2"},"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"}