{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T10:55:15Z","timestamp":1772880915882,"version":"3.50.1"},"reference-count":25,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We consider the 3D stochastic Navier\u2013Stokes equation on the torus.\nOur main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution.\nWe prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1\/2 in time.\nThe result holds up to the possible blow-up of the (time-discrete) solution.\nOur approach is based on discrete stopping times for the (time-discrete) solution.<\/jats:p>","DOI":"10.1515\/cmam-2023-0052","type":"journal-article","created":{"date-parts":[[2023,8,7]],"date-time":"2023-08-07T15:41:35Z","timestamp":1691422895000},"page":"577-597","source":"Crossref","is-referenced-by-count":4,"title":["Space-Time Approximation of Local Strong Solutions to the 3D Stochastic Navier\u2013Stokes Equations"],"prefix":"10.1515","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2787-3250","authenticated-orcid":false,"given":"Dominic","family":"Breit","sequence":"first","affiliation":[{"name":"Institute of Mathematics , TU Clausthal , Erzstra\u00dfe 1, 38678 Clausthal-Zellerfeld , Germany ; and Department of Mathematics, Heriot-Watt University, Riccarton Edinburgh EH14 4AS, United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alan","family":"Dodgson","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Heriot-Watt University , Riccarton Edinburgh EH14 4AS , United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,8,8]]},"reference":[{"key":"2024070116104937691_j_cmam-2023-0052_ref_001","doi-asserted-by":"crossref","unstructured":"A. Bensoussan and J. Frehse,\nLocal solutions for stochastic Navier Stokes equations,\nM2AN Math. Model. Numer. Anal. 34 (2000), 241\u2013273.","DOI":"10.1051\/m2an:2000140"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_002","doi-asserted-by":"crossref","unstructured":"A. Bensoussan and R. Temam,\n\u00c9quations stochastiques du type Navier\u2013Stokes,\nJ. Funct. Anal. 13 (1973), 195\u2013222.","DOI":"10.1016\/0022-1236(73)90045-1"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_003","doi-asserted-by":"crossref","unstructured":"H. Bessaih and A. Millet,\nStrong \n                  \n                     \n                        \n                           L\n                           2\n                        \n                     \n                     \n                     L^{2}\n                  \n                convergence of time numerical schemes for the stochastic two-dimensional Navier\u2013Stokes equations,\nIMA J. Numer. Anal. 39 (2019), no. 4, 2135\u20132167.","DOI":"10.1093\/imanum\/dry058"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_004","doi-asserted-by":"crossref","unstructured":"H. Bessaih and A. Millet,\nStrong rates of convergence of space-time discretization schemes for the 2D Navier\u2013Stokes equations with additive noise,\nStoch. Dyn. 22 (2022), no. 2, Paper No. 2240005.","DOI":"10.1142\/S0219493722400056"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_005","doi-asserted-by":"crossref","unstructured":"D. Breit,\nExistence theory for stochastic power law fluids,\nJ. Math. Fluid Mech. 17 (2015), no. 2, 295\u2013326.","DOI":"10.1007\/s00021-015-0203-z"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_006","doi-asserted-by":"crossref","unstructured":"D. Breit,\nExistence Theory for Generalized Newtonian Fluids,\nMath. Sci. Eng.,\nElsevier\/Academic, London, 2017.","DOI":"10.1090\/conm\/666\/13242"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_007","doi-asserted-by":"crossref","unstructured":"D. Breit and A. Dodgson,\nConvergence rates for the numerical approximation of the 2D stochastic Navier\u2013Stokes equations,\nNumer. Math. 147 (2021), 553\u2013578.","DOI":"10.1007\/s00211-021-01181-z"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_008","doi-asserted-by":"crossref","unstructured":"D. Breit, E. Feireisl and M. Hofmanov\u00e1,\nOn solvability and ill-posedness of the compressible Euler system subject to stochastic forces,\nAnal. PDE 13 (2020), 371\u2013402.","DOI":"10.2140\/apde.2020.13.371"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_009","doi-asserted-by":"crossref","unstructured":"D. Breit and A. Prohl,\nError analysis for 2D stochastic Navier\u2013Stokes equations in bounded domains with Dirichlet data,\npreprint 2022, https:\/\/arxiv.org\/abs\/2109.06495v2;\nto apper in Found. Comp. Math.","DOI":"10.1007\/s10208-023-09621-y"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_010","doi-asserted-by":"crossref","unstructured":"D. Breit and A. Prohl,\nNumerical analysis of 2D Navier\u2013Stokes equations with additive stochastic forcing,\nIMA J. Numer. Anal. 43 (2023), 1391\u20131421.","DOI":"10.1093\/imanum\/drac023"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_011","doi-asserted-by":"crossref","unstructured":"Z. Brze\u017aniak, E. Carelli and A. Prohl,\nFinite-element-based discretizations of the incompressible Navier\u2013Stokes equations with multiplicative random forcing,\nIMA J. Numer. Anal. 33 (2013), no. 3, 771\u2013824.","DOI":"10.1093\/imanum\/drs032"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_012","unstructured":"Z. Brze\u017aniak and S. Peszat,\nStrong local and global solutions for stochastic Navier\u2013Stokes equations,\nInfinite Dimensional Stochastic Analysis (Amsterdam 1999),\nVerh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. 52,\nRoyal Netherlands Academy of Arts and Sciences, Amsterdam (2000), 85\u201398."},{"key":"2024070116104937691_j_cmam-2023-0052_ref_013","doi-asserted-by":"crossref","unstructured":"E. Carelli and A. Prohl,\nRates of convergence for discretizations of the stochastic incompressible Navier\u2013Stokes equations,\nSIAM J. Numer. Anal. 50 (2012), no. 5, 2467\u20132496.","DOI":"10.1137\/110845008"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_014","doi-asserted-by":"crossref","unstructured":"G. Da Prato and J. Zabczyk,\nStochastic Equations in Infinite Dimensions,\nEncyclopedia Math. Appl. 44,\nCambridge University, Cambridge, 1992.","DOI":"10.1017\/CBO9780511666223"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_015","doi-asserted-by":"crossref","unstructured":"F. Flandoli,\nAn introduction to 3D stochastic fluid dynamics,\nSPDE in Hydrodynamic: Recent Progress and Prospects,\nLecture Notes in Math. 1942,\nSpringer, Berlin (2008), 51\u2013150.","DOI":"10.1007\/978-3-540-78493-7_2"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_016","doi-asserted-by":"crossref","unstructured":"F. Flandoli and D. Ga\u0327tarek,\nMartingale and stationary solutions for stochastic Navier\u2013Stokes equations,\nProbab. Theory Related Fields 102 (1995), no. 3, 367\u2013391.","DOI":"10.1007\/BF01192467"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_017","doi-asserted-by":"crossref","unstructured":"F. Flandoli and D. Luo,\nHigh mode transport noise improves vorticity blow-up control in 3D Navier\u2013Stokes equations,\nProbab. Theory Related Fields 180 (2021), 309\u2013363.","DOI":"10.1007\/s00440-021-01037-5"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_018","doi-asserted-by":"crossref","unstructured":"V. Girault and P.-A. Raviart,\nFinite Element Methods for Navier\u2013Stokes Equations,\nSpringer Ser. Comput. Math. 5,\nSpringer, Berlin, 1986.","DOI":"10.1007\/978-3-642-61623-5"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_019","doi-asserted-by":"crossref","unstructured":"N. Glatt-Holtz and M. Ziane,\nStrong pathwise solutions of the stochastic Navier\u2013Stokes system,\nAdv. Differential Equations 14 (2009), no. 5\u20136, 567\u2013600.","DOI":"10.57262\/ade\/1355867260"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_020","doi-asserted-by":"crossref","unstructured":"J. G. Heywood and R. Rannacher,\nFinite element approximation of the nonstationary Navier\u2013Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization,\nSIAM J. Numer. Anal. 19 (1982), 275\u2013311.","DOI":"10.1137\/0719018"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_021","doi-asserted-by":"crossref","unstructured":"M. Hofmanov\u00e1, R. Zhu and X. Zhu,\nNon-uniqueness in law of stochastic 3D Navier\u2013Stokes equations,\nJ. Eur. Math. Soc. (2023), 10.4171\/JEMS\/1360.","DOI":"10.4171\/jems\/1360"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_022","doi-asserted-by":"crossref","unstructured":"M. Hofmanov\u00e1, R. Zhu and X. Zhu,\nOn ill- and well-posedness of dissipative martingale solutions to stochastic 3D Euler equations,\nComm. Pure Appl. Math. 75 (2022), no. 11, 2446\u20132510.","DOI":"10.1002\/cpa.22023"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_023","doi-asserted-by":"crossref","unstructured":"J. U. Kim,\nStrong solutions of the stochastic Navier\u2013Stokes equations in \n                  \n                     \n                        \n                           R\n                           3\n                        \n                     \n                     \n                     \\mathbb{R}^{3}\n                  \n               ,\nIndiana Univ. Math. J. 59 (2010), no. 4, 1417\u20131450.","DOI":"10.1512\/iumj.2010.59.3930"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_024","doi-asserted-by":"crossref","unstructured":"R. Mikulevicius,\nOn strong \n                  \n                     \n                        \n                           H\n                           2\n                           1\n                        \n                     \n                     \n                     H_{2}^{1}\n                  \n               -solutions of stochastic Navier\u2013Stokes equation in a bounded domain,\nSIAM J. Math. Anal. 41 (2009), no. 3, 1206\u20131230.","DOI":"10.1137\/0807433747"},{"key":"2024070116104937691_j_cmam-2023-0052_ref_025","doi-asserted-by":"crossref","unstructured":"M. Romito,\nSome probabilistic topics in the Navier-Stokes equations,\nRecent Progress in the Theory of the Euler and Navier\u2013Stokes Equations,\nLondon Math. Soc. Lecture Note Ser. 430,\nCambridge University, Cambridge (2016), 175\u2013232.","DOI":"10.1017\/CBO9781316407103.011"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0052\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0052\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T16:13:23Z","timestamp":1719850403000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0052\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,8]]},"references-count":25,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,6,26]]},"published-print":{"date-parts":[[2024,7,1]]}},"alternative-id":["10.1515\/cmam-2023-0052"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0052","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,8]]}}}