{"id":"https://openalex.org/W2032229183","doi":"https://doi.org/10.1137/s0036141002409167","title":"Stochastic Navier--Stokes Equations for Turbulent Flows","display_name":"Stochastic Navier--Stokes Equations for Turbulent Flows","publication_year":2004,"publication_date":"2004-01-01","ids":{"openalex":"https://openalex.org/W2032229183","doi":"https://doi.org/10.1137/s0036141002409167","mag":"2032229183"},"language":"en","primary_location":{"id":"doi:10.1137/s0036141002409167","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036141002409167","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5085482977","display_name":"R. Mikulev\u00ed\u010dius","orcid":"https://orcid.org/0000-0002-7035-0191"},"institutions":[{"id":"https://openalex.org/I1174212","display_name":"University of Southern California","ror":"https://ror.org/03taz7m60","country_code":"US","type":"education","lineage":["https://openalex.org/I1174212"]},{"id":"https://openalex.org/I2800817003","display_name":"California Southern University","ror":"https://ror.org/058zz0t50","country_code":"US","type":"education","lineage":["https://openalex.org/I2800817003"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"R. Mikulevicius","raw_affiliation_strings":["University of Southern California"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Southern California","institution_ids":["https://openalex.org/I2800817003","https://openalex.org/I1174212"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5044537548","display_name":"B. L. Rozovski\u012d","orcid":null},"institutions":[{"id":"https://openalex.org/I1174212","display_name":"University of Southern California","ror":"https://ror.org/03taz7m60","country_code":"US","type":"education","lineage":["https://openalex.org/I1174212"]},{"id":"https://openalex.org/I2800817003","display_name":"California Southern University","ror":"https://ror.org/058zz0t50","country_code":"US","type":"education","lineage":["https://openalex.org/I2800817003"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"B. L. Rozovskii","raw_affiliation_strings":["University of Southern California"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Southern California","institution_ids":["https://openalex.org/I2800817003","https://openalex.org/I1174212"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":8.866,"has_fulltext":false,"cited_by_count":298,"citation_normalized_percentile":{"value":0.9762281,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"35","issue":"5","first_page":"1250","last_page":"1310"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10940","display_name":"Navier-Stokes equation solutions","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.986299991607666,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.7492160797119141},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6477696895599365},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.5198041200637817},{"id":"https://openalex.org/keywords/turbulence","display_name":"Turbulence","score":0.4910428524017334},{"id":"https://openalex.org/keywords/navier\u2013stokes-equations","display_name":"Navier\u2013Stokes equations","score":0.4853323996067047},{"id":"https://openalex.org/keywords/euler-equations","display_name":"Euler equations","score":0.4671596586704254},{"id":"https://openalex.org/keywords/flow","display_name":"Flow (mathematics)","score":0.4495212435722351},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.41605526208877563},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.33281105756759644},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.17105931043624878},{"id":"https://openalex.org/keywords/compressibility","display_name":"Compressibility","score":0.10953876376152039}],"concepts":[{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.7492160797119141},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6477696895599365},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.5198041200637817},{"id":"https://openalex.org/C196558001","wikidata":"https://www.wikidata.org/wiki/Q190132","display_name":"Turbulence","level":2,"score":0.4910428524017334},{"id":"https://openalex.org/C2781278361","wikidata":"https://www.wikidata.org/wiki/Q201321","display_name":"Navier\u2013Stokes equations","level":3,"score":0.4853323996067047},{"id":"https://openalex.org/C38409319","wikidata":"https://www.wikidata.org/wiki/Q375175","display_name":"Euler equations","level":2,"score":0.4671596586704254},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.4495212435722351},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.41605526208877563},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.33281105756759644},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.17105931043624878},{"id":"https://openalex.org/C84655787","wikidata":"https://www.wikidata.org/wiki/Q8067817","display_name":"Compressibility","level":2,"score":0.10953876376152039},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/s0036141002409167","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036141002409167","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.161.4144","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.161.4144","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://ima.umn.edu/prob-pde/reprints-preprints/rozovskii/NSESIAM7IMA.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":58,"referenced_works":["https://openalex.org/W205656172","https://openalex.org/W221422793","https://openalex.org/W221832730","https://openalex.org/W366920335","https://openalex.org/W581771450","https://openalex.org/W1488877410","https://openalex.org/W1526194873","https://openalex.org/W1542004730","https://openalex.org/W1547825328","https://openalex.org/W1594572592","https://openalex.org/W1608564739","https://openalex.org/W1609336153","https://openalex.org/W1631291176","https://openalex.org/W1634277530","https://openalex.org/W1971773540","https://openalex.org/W1973771388","https://openalex.org/W1974264121","https://openalex.org/W1976512063","https://openalex.org/W1984533458","https://openalex.org/W1992168277","https://openalex.org/W1992254982","https://openalex.org/W1993988025","https://openalex.org/W1994088960","https://openalex.org/W1996949039","https://openalex.org/W1998107368","https://openalex.org/W1998319262","https://openalex.org/W2000315008","https://openalex.org/W2005780385","https://openalex.org/W2011032774","https://openalex.org/W2015535719","https://openalex.org/W2018717342","https://openalex.org/W2021841897","https://openalex.org/W2025578652","https://openalex.org/W2037777951","https://openalex.org/W2040452482","https://openalex.org/W2049831275","https://openalex.org/W2055105988","https://openalex.org/W2059091656","https://openalex.org/W2087164036","https://openalex.org/W2087895654","https://openalex.org/W2156345340","https://openalex.org/W2167720109","https://openalex.org/W2178464435","https://openalex.org/W2314913127","https://openalex.org/W2332771484","https://openalex.org/W2334723427","https://openalex.org/W2506877196","https://openalex.org/W2990787487","https://openalex.org/W3021025175","https://openalex.org/W4213021285","https://openalex.org/W4232411235","https://openalex.org/W4237024386","https://openalex.org/W4238243565","https://openalex.org/W4245767272","https://openalex.org/W4247047207","https://openalex.org/W4285719527","https://openalex.org/W4292079233","https://openalex.org/W4300475775"],"related_works":["https://openalex.org/W4294029204","https://openalex.org/W1514935853","https://openalex.org/W2042809915","https://openalex.org/W650993247","https://openalex.org/W3198045404","https://openalex.org/W2038540943","https://openalex.org/W3180958935","https://openalex.org/W1984672658","https://openalex.org/W278619367","https://openalex.org/W2056077850"],"abstract_inverted_index":{"This":[0],"paper":[1],"concerns":[2],"the":[3,8,31,37,49,65,72,79,89,99,103,107,117,127,130,142,148,154,165],"fluid":[4,52,59],"dynamics":[5],"modelled":[6],"by":[7,36,153],"stochastic":[9,118,155],"flow":[10],"\\left\\{":[11],"\\begin{array}{l}":[12],"\\boldsymbol{\\dot{\\eta}}\\left(":[13],"t,x\\right)":[14,18,22],"=\\boldsymbol{u}\\left(":[15],"t,\\boldsymbol{\\eta}":[16],"\\left(":[17],"\\right)":[19,23],"+\\boldsymbol{\\sigma}\\left(":[20],"t,\\boldsymbol{\\eta}\\left(":[21],"\\circ\\dot{W},":[24],"\\\\":[25,26],"\\boldsymbol{\\eta}(0,x)=x,":[27],"\\end{array}":[28],"\\right.":[29],"where":[30],"turbulent":[32,55],"term":[33],"is":[34,46,139],"driven":[35],"white":[38],"noise":[39],"$\\dot{W}$.":[40],"The":[41,82],"motivation":[42],"for":[43,64],"this":[44],"setting":[45],"to":[47,116,161],"understand":[48],"motion":[50],"of":[51,71,95,102,111,134,150,164],"parcels":[53],"in":[54,121],"and":[56,69,91,109,132,158],"randomly":[57,92],"forced":[58,93],"flows.":[60],"Stochastic":[61],"Euler":[62],"equations":[63,84],"undetermined":[66],"components":[67],"$\\boldsymbol{u}(t,x)$":[68],"$\\boldsymbol{\\sigma}(t,x)$":[70],"spatial":[73],"velocity":[74],"field":[75],"are":[76],"derived":[77],"from":[78],"first":[80],"principles.":[81],"resulting":[83],"include":[85],"as":[86],"particular":[87],"cases":[88],"deterministic":[90],"counterparts":[94],"these":[96],"equations.":[97],"In":[98,126,141],"second":[100],"part":[101],"paper,":[104],"we":[105,145],"prove":[106],"existence":[108,131],"uniqueness":[110,133],"a":[112,135],"strong":[113,137],"local":[114],"solution":[115,138],"Navier--Stokes":[119,156],"equation":[120,157],"$W_{p}^{1}(\\boldsymbol{R}^{d}),d":[122],">1,p":[123],">":[124],"d.":[125],"two-dimensional":[128],"case,":[129],"global":[136],"shown.":[140],"third":[143],"part,":[144],"deal":[146],"with":[147],"propagation":[149],"Wiener":[151],"chaos":[152],"its":[159],"relation":[160],"statistical":[162],"moments":[163],"solution.":[166]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":21},{"year":2024,"cited_by_count":16},{"year":2023,"cited_by_count":21},{"year":2022,"cited_by_count":16},{"year":2021,"cited_by_count":20},{"year":2020,"cited_by_count":21},{"year":2019,"cited_by_count":13},{"year":2018,"cited_by_count":16},{"year":2017,"cited_by_count":15},{"year":2016,"cited_by_count":12},{"year":2015,"cited_by_count":15},{"year":2014,"cited_by_count":17},{"year":2013,"cited_by_count":13},{"year":2012,"cited_by_count":17}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
