{"id":"https://openalex.org/W7207744657","doi":"https://doi.org/10.48550/arxiv.2609.02769","title":"Divergence-Free Approximation in Sobolev and Lebesgue Spaces on General Unbounded Domains with Applications to Energy Equality in Fluid Dynamics","display_name":"Divergence-Free Approximation in Sobolev and Lebesgue Spaces on General Unbounded Domains with Applications to Energy Equality in Fluid Dynamics","publication_year":2026,"publication_date":"2026-09-02","ids":{"openalex":"https://openalex.org/W7207744657","doi":"https://doi.org/10.48550/arxiv.2609.02769"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.02769","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02769","pdf_url":"https://arxiv.org/pdf/2609.02769","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.02769","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5149477758","display_name":"Akram Khan","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I154851008","display_name":"Indian Institute of Technology Roorkee","ror":"https://ror.org/00582g326","country_code":"IN","type":"education","lineage":["https://openalex.org/I154851008"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Khan, Akram","raw_affiliation_strings":["Department of Mathematics , Indian Institute of Technology Roorkee-IIT Roorkee , Haridwar High- way , Roorkee , Uttarakhand 247667 , INDIA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Indian Institute of Technology Roorkee-IIT Roorkee , Haridwar High- way , Roorkee , Uttarakhand 247667 , INDIA","institution_ids":["https://openalex.org/I154851008"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5153458196","display_name":"Sagar Gautam","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I154851008","display_name":"Indian Institute of Technology Roorkee","ror":"https://ror.org/00582g326","country_code":"IN","type":"education","lineage":["https://openalex.org/I154851008"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Gautam, Sagar","raw_affiliation_strings":["Department of Mathematics , Indian Institute of Technology Roorkee-IIT Roorkee , Haridwar High- way , Roorkee , Uttarakhand 247667 , INDIA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Indian Institute of Technology Roorkee-IIT Roorkee , Haridwar High- way , Roorkee , Uttarakhand 247667 , INDIA","institution_ids":["https://openalex.org/I154851008"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5123506675","display_name":"Manil T. Mohan","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I154851008","display_name":"Indian Institute of Technology Roorkee","ror":"https://ror.org/00582g326","country_code":"IN","type":"education","lineage":["https://openalex.org/I154851008"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Mohan, Manil T.","raw_affiliation_strings":["Department of Mathematics , Indian Institute of Technology Roorkee-IIT Roorkee , Haridwar High- way , Roorkee , Uttarakhand 247667 , INDIA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Indian Institute of Technology Roorkee-IIT Roorkee , Haridwar High- way , Roorkee , Uttarakhand 247667 , INDIA","institution_ids":["https://openalex.org/I154851008"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I154851008"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10940","display_name":"Navier-Stokes equation solutions","score":0.9907000064849854,"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":0.9907000064849854,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.00279999990016222,"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/T11049","display_name":"Advanced Harmonic Analysis Research","score":0.0017000000225380063,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/unbounded-domains","display_name":"unbounded domains","score":0.8809999823570251},{"id":"https://openalex.org/keywords/sobolev-spaces","display_name":"Sobolev spaces","score":0.7850000262260437},{"id":"https://openalex.org/keywords/lp-spaces","display_name":"Lp spaces","score":0.9919999837875366},{"id":"https://openalex.org/keywords/simultaneous-approximation","display_name":"simultaneous approximation","score":0.8100000023841858},{"id":"https://openalex.org/keywords/energy-equality","display_name":"energy equality","score":0.9570000171661377},{"id":"https://openalex.org/keywords/leray-hopf-weak-solutions","display_name":"Leray-Hopf weak solutions","score":0.9570000171661377}],"concepts":[{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.7717000246047974},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7714999914169312},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6018999814987183},{"id":"https://openalex.org/C196267783","wikidata":"https://www.wikidata.org/wiki/Q4392490","display_name":"Resolvent","level":2,"score":0.5909000039100647},{"id":"https://openalex.org/C14158598","wikidata":"https://www.wikidata.org/wiki/Q853831","display_name":"Lebesgue integration","level":2,"score":0.4860999882221222},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.48330000042915344},{"id":"https://openalex.org/C172294467","wikidata":"https://www.wikidata.org/wiki/Q305936","display_name":"Lp space","level":3,"score":0.4438999891281128},{"id":"https://openalex.org/C12871628","wikidata":"https://www.wikidata.org/wiki/Q7598372","display_name":"Standard probability space","level":2,"score":0.42329999804496765},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.36410000920295715},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.36230000853538513},{"id":"https://openalex.org/C194571181","wikidata":"https://www.wikidata.org/wiki/Q7449306","display_name":"Semi-infinite","level":2,"score":0.3578000068664551},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.34450000524520874},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.34369999170303345},{"id":"https://openalex.org/C152306953","wikidata":"https://www.wikidata.org/wiki/Q621744","display_name":"Approximation property","level":3,"score":0.34139999747276306},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.3330000042915344},{"id":"https://openalex.org/C84655787","wikidata":"https://www.wikidata.org/wiki/Q8067817","display_name":"Compressibility","level":2,"score":0.33079999685287476},{"id":"https://openalex.org/C145242015","wikidata":"https://www.wikidata.org/wiki/Q774123","display_name":"Approximation theory","level":2,"score":0.31470000743865967},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.2928999960422516},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28679999709129333},{"id":"https://openalex.org/C193386753","wikidata":"https://www.wikidata.org/wiki/Q1130396","display_name":"Approximations of \u03c0","level":2,"score":0.26969999074935913},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.26829999685287476},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.2522999942302704},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.251800000667572}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.02769","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02769","pdf_url":"https://arxiv.org/pdf/2609.02769","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.02769","doi":"https://doi.org/10.48550/arxiv.2609.02769","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.02769","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.02769","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02769","pdf_url":"https://arxiv.org/pdf/2609.02769","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G4916653689","display_name":null,"funder_award_id":"02011/13/2025/NBHM(R.P)/R&amp;D II/1137","funder_id":"https://openalex.org/F4320320768","funder_display_name":"Department of Atomic Energy, Government of India"}],"funders":[{"id":"https://openalex.org/F4320320721","display_name":"Council of Scientific and Industrial Research, India","ror":"https://ror.org/021wm7p51"},{"id":"https://openalex.org/F4320320768","display_name":"Department of Atomic Energy, Government of India","ror":"https://ror.org/02m388s04"},{"id":"https://openalex.org/F4320320873","display_name":"Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg","ror":"https://ror.org/00f7hpc57"},{"id":"https://openalex.org/F4320322724","display_name":"Ministry of Education, India","ror":"https://ror.org/048xjjh50"},{"id":"https://openalex.org/F4320323196","display_name":"National Board for Higher Mathematics","ror":"https://ror.org/02m388s04"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7207744657.pdf","grobid_xml":"https://content.openalex.org/works/W7207744657.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2929425553","https://openalex.org/W4403746920","https://openalex.org/W2809840466","https://openalex.org/W2041507510","https://openalex.org/W35960826","https://openalex.org/W4386016646","https://openalex.org/W2114570393","https://openalex.org/W1969960146","https://openalex.org/W2147315827","https://openalex.org/W7147662935"],"abstract_inverted_index":{"We":[0,134],"construct":[1,42],"divergence-free,":[2],"vector-valued":[3],"approximation":[4,48,58,132,158],"functions":[5],"on":[6,16,94,174],"the":[7,34,47,56,88,99,110,115,141,156,171],"whole":[8],"space":[9],"$\\mathbb{R}^d$,":[10,95],"$d\\geq":[11],"2$,":[12],"as":[13,15],"well":[14],"general":[17,105,175],"unbounded":[18,106,161,176],"domains":[19,54],"of":[20,87,140,147,170],"uniform":[21],"$\\mathrm{C}^{1,1}$-type.":[22],"These":[23],"approximations":[24],"converge":[25],"simultaneously":[26],"in":[27],"both":[28,98],"Sobolev":[29],"and":[30,55,64,101,122],"Lebesgue":[31],"spaces.":[32],"In":[33],"whole-space":[35],"setting,":[36],"we":[37,79,108,163],"employ":[38,109],"Bogovski\\u\u0131\\":[39],"operators":[40],"to":[41,129],"such":[43],"approximations,":[44],"thereby":[45],"extending":[46],"theory":[49],"developed":[50,118],"for":[51,83,160,167],"smooth":[52],"bounded":[53],"simultaneous":[57,131,157],"framework":[59,159],"introduced":[60],"by":[61,119,151],"\\emph{Fefferman,":[62],"Hajduk,":[63],"Robinson,":[65],"{Proc.":[66],"Lond.":[67],"Math.":[68],"Soc.}":[69],"(3)":[70],"\\":[71],"{125}":[72],"(2022),":[73],"no.~4,":[74],"759-777}.":[75],"As":[76],"an":[77],"application,":[78],"establish":[80,136,164],"energy":[81,165],"equality":[82,166],"Leray-Hopf":[84],"weak":[85,168],"solutions":[86,169],"incompressible":[89],"convective":[90],"Brinkman-Forchheimer":[91],"(CBF)":[92],"equations":[93,173],"$d\\in\\{2,3\\}$":[96],"covering":[97],"critical":[100],"supercritical":[102],"regimes.":[103],"For":[104],"domains,":[107,162],"resolvent":[111],"operator":[112],"associated":[113],"with":[114,155],"Stokes":[116],"operator,":[117],"\\emph{Farwig,":[120],"Kozono":[121],"Sohr,":[123],"{Acta":[124],"Math.},":[125],"{195}":[126],"(2005),":[127],"21-53},":[128],"obtain":[130],"results.":[133],"also":[135],"a":[137],"generalized":[138],"version":[139],"classical":[142],"Lions-Magenes":[143],"lemma,":[144],"which":[145],"is":[146],"independent":[148],"interest.":[149],"Finally,":[150],"combining":[152],"this":[153],"result":[154],"CBF":[172],"domains.":[177]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-04T00:00:00"}
