{"id":"https://openalex.org/W2005837188","doi":"https://doi.org/10.1137/s0036141098346177","title":"Pointwise Decay of Solutions and of Higher Derivatives to Navier--Stokes Equations","display_name":"Pointwise Decay of Solutions and of Higher Derivatives to Navier--Stokes Equations","publication_year":2000,"publication_date":"2000-01-01","ids":{"openalex":"https://openalex.org/W2005837188","doi":"https://doi.org/10.1137/s0036141098346177","mag":"2005837188"},"language":"en","primary_location":{"id":"doi:10.1137/s0036141098346177","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036141098346177","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/A5001731799","display_name":"Ch\u00e9rif Amrouche","orcid":"https://orcid.org/0000-0002-9175-3555"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cherif Amrouche","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026303492","display_name":"Vivette Girault","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Vivette Girault","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040714345","display_name":"Mar\u00eda E. Schonbek","orcid":"https://orcid.org/0000-0002-9917-8495"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Maria Elena Schonbek","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5038031859","display_name":"Tomas Schonbek","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tomas P. Schonbek","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.9458,"has_fulltext":false,"cited_by_count":45,"citation_normalized_percentile":{"value":0.89156936,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"31","issue":"4","first_page":"740","last_page":"753"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10940","display_name":"Navier-Stokes equation solutions","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T10940","display_name":"Navier-Stokes equation solutions","score":0.9998000264167786,"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/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.994700014591217,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9842000007629395,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/pointwise","display_name":"Pointwise","score":0.9388439655303955},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7642024755477905},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.6735958456993103},{"id":"https://openalex.org/keywords/navier\u2013stokes-equations","display_name":"Navier\u2013Stokes equations","score":0.6574031114578247},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.5738484263420105},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5066016912460327},{"id":"https://openalex.org/keywords/compressibility","display_name":"Compressibility","score":0.4809683859348297},{"id":"https://openalex.org/keywords/moment","display_name":"Moment (physics)","score":0.4647487998008728},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.42872804403305054},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3095409870147705},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1801331639289856},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.07734012603759766},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.0713362991809845}],"concepts":[{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.9388439655303955},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7642024755477905},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6735958456993103},{"id":"https://openalex.org/C2781278361","wikidata":"https://www.wikidata.org/wiki/Q201321","display_name":"Navier\u2013Stokes equations","level":3,"score":0.6574031114578247},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5738484263420105},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5066016912460327},{"id":"https://openalex.org/C84655787","wikidata":"https://www.wikidata.org/wiki/Q8067817","display_name":"Compressibility","level":2,"score":0.4809683859348297},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.4647487998008728},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.42872804403305054},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3095409870147705},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1801331639289856},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.07734012603759766},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0713362991809845},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/s0036141098346177","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036141098346177","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:aleph.bib-bvb.de:BVB01-008360493","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:kluedo.ub.rptu.de:832","is_oa":false,"landing_page_url":"https://kluedo.ub.rptu.de/frontdoor/index/index/docId/832","pdf_url":null,"source":{"id":"https://openalex.org/S4306402046","display_name":"Publication Server of Kaiserslautern University of Technology (Kaiserslautern University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I153267046","host_organization_name":"University of Kaiserslautern","host_organization_lineage":["https://openalex.org/I153267046"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"preprint"},{"id":"pmh:oai:kluedo.ub.uni-kl.de:832","is_oa":false,"landing_page_url":"https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/832","pdf_url":null,"source":{"id":"https://openalex.org/S4306402046","display_name":"Publication Server of Kaiserslautern University of Technology (Kaiserslautern University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I153267046","host_organization_name":"University of Kaiserslautern","host_organization_lineage":["https://openalex.org/I153267046"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"preprint"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W1488877410","https://openalex.org/W1584610719","https://openalex.org/W1982418511","https://openalex.org/W1985765218","https://openalex.org/W2001691824","https://openalex.org/W2002508570","https://openalex.org/W2021073558","https://openalex.org/W2021964921","https://openalex.org/W2031343718","https://openalex.org/W2044196646","https://openalex.org/W2059615432","https://openalex.org/W2067971241","https://openalex.org/W2069367520","https://openalex.org/W2086992944","https://openalex.org/W2090043039","https://openalex.org/W2488440663","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2140020064","https://openalex.org/W143502885","https://openalex.org/W2103468410","https://openalex.org/W1856228368","https://openalex.org/W1968809652","https://openalex.org/W3197542402","https://openalex.org/W2971527398","https://openalex.org/W2086538707","https://openalex.org/W1025150868","https://openalex.org/W2241171993"],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3,29],"study":[4],"the":[5,10,16,35,43,51],"space-time":[6,54],"asymptotic":[7],"behavior":[8],"of":[9,45],"solutions,":[11],"and":[12,79],"their":[13],"derivatives,":[14],"to":[15,34],"incompressible":[17],"Navier--Stokes":[18,36],"equations":[19,37],"in":[20,40],"dimension":[21],"$2\\le":[22],"n":[23],"\\le":[24,59],"5$.":[25],"Using":[26],"moment":[27],"estimates":[28],"obtain":[30],"that":[31],"strong":[32],"solutions":[33],"which":[38],"decay":[39],"L2":[41],"at":[42],"rate":[44],"$\\|u(t)\\|_2":[46],"\\leq":[47,63],"C(t+1)^{-\\mu}$":[48],"will":[49],"have":[50],"following":[52],"pointwise":[53],"decay,":[55],"for":[56],"$0\\le":[57],"k":[58],"n/2$:":[60],"$$":[61,66],"|D^{\\a}u(x,t)|":[62],"C_{k,m}":[64],"\\frac{1}{(t+1)^{\\rho_0}(1+|x|^2)^{k/2}},":[65],"\\noindent":[67],"where":[68],"$\\rho_O":[69],"=":[70],"(1-2k/n)(":[71],"m/2":[72],"+":[73],"\\mu":[74],"+n/4)$,":[75],"$|\\a":[76],"|=":[77],"m$":[78],"$\\mu":[80],">":[81],"\\frac{n}{4}$.":[82]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2021,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}