{"id":"https://openalex.org/W2692960743","doi":"https://doi.org/10.1137/16m1063927","title":"Viscous Scalar Rarefaction Waves","display_name":"Viscous Scalar Rarefaction Waves","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2692960743","doi":"https://doi.org/10.1137/16m1063927","mag":"2692960743"},"language":"en","primary_location":{"id":"doi:10.1137/16m1063927","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1063927","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/A5050124790","display_name":"Tai-Ping Liu","orcid":"https://orcid.org/0000-0002-3843-8855"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tai-Ping Liu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5085232405","display_name":"Haitao Wang","orcid":"https://orcid.org/0000-0002-7889-4206"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Haitao Wang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.11751962,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"49","issue":"3","first_page":"2061","last_page":"2100"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10940","display_name":"Navier-Stokes equation solutions","score":0.9997000098228455,"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":0.9997000098228455,"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":0.9994000196456909,"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"}},{"id":"https://openalex.org/T11012","display_name":"Gas Dynamics and Kinetic Theory","score":0.9990000128746033,"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/mathematics","display_name":"Mathematics","score":0.7881195545196533},{"id":"https://openalex.org/keywords/conservation-law","display_name":"Conservation law","score":0.7034247517585754},{"id":"https://openalex.org/keywords/rarefaction","display_name":"Rarefaction (ecology)","score":0.6170973777770996},{"id":"https://openalex.org/keywords/pointwise","display_name":"Pointwise","score":0.6122494339942932},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.6017663478851318},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5948098301887512},{"id":"https://openalex.org/keywords/burgers-equation","display_name":"Burgers' equation","score":0.5709136128425598},{"id":"https://openalex.org/keywords/riemann-problem","display_name":"Riemann problem","score":0.5567415356636047},{"id":"https://openalex.org/keywords/riemann-hypothesis","display_name":"Riemann hypothesis","score":0.5139877200126648},{"id":"https://openalex.org/keywords/transformation","display_name":"Transformation (genetics)","score":0.44704392552375793},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.399375319480896},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.14032688736915588}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7881195545196533},{"id":"https://openalex.org/C3445786","wikidata":"https://www.wikidata.org/wiki/Q205805","display_name":"Conservation law","level":2,"score":0.7034247517585754},{"id":"https://openalex.org/C132705693","wikidata":"https://www.wikidata.org/wiki/Q7294543","display_name":"Rarefaction (ecology)","level":3,"score":0.6170973777770996},{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.6122494339942932},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.6017663478851318},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5948098301887512},{"id":"https://openalex.org/C129747778","wikidata":"https://www.wikidata.org/wiki/Q1014918","display_name":"Burgers' equation","level":3,"score":0.5709136128425598},{"id":"https://openalex.org/C40709475","wikidata":"https://www.wikidata.org/wiki/Q2152221","display_name":"Riemann problem","level":3,"score":0.5567415356636047},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.5139877200126648},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.44704392552375793},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.399375319480896},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.14032688736915588},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C62648534","wikidata":"https://www.wikidata.org/wiki/Q706937","display_name":"Species diversity","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/16m1063927","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1063927","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"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Life in Land","id":"https://metadata.un.org/sdg/15","score":0.699999988079071}],"awards":[{"id":"https://openalex.org/G4392610590","display_name":null,"funder_award_id":"DMS-0709248","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321041","display_name":"Academia Sinica","ror":"https://ror.org/05bxb3784"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":9,"referenced_works":["https://openalex.org/W1983267046","https://openalex.org/W1993760926","https://openalex.org/W2035158114","https://openalex.org/W2055861004","https://openalex.org/W2080284917","https://openalex.org/W2154955497","https://openalex.org/W2182320420","https://openalex.org/W2186813477","https://openalex.org/W4249384777"],"related_works":["https://openalex.org/W2800583016","https://openalex.org/W2480601066","https://openalex.org/W1992723975","https://openalex.org/W1979469089","https://openalex.org/W2902100385","https://openalex.org/W3174853118","https://openalex.org/W3085759807","https://openalex.org/W2921770740","https://openalex.org/W2083233781","https://openalex.org/W4387025432"],"abstract_inverted_index":{"We":[0,34],"study":[1],"the":[2,17,20,28,31,36,44,48],"pointwise":[3],"behavior":[4,26],"of":[5,19,27,30,43],"rarefaction":[6],"waves":[7],"for":[8,47],"scalar":[9],"viscous":[10],"conservation":[11],"laws.":[12],"Our":[13],"primary":[14],"goal":[15],"is":[16],"analysis":[18],"initial":[21],"layer":[22],"and":[23],"large":[24],"time":[25],"solution":[29],"Riemann":[32],"problem.":[33],"use":[35,42],"Green's":[37],"function":[38],"approach,":[39],"making":[40],"essential":[41],"Hopf--Cole":[45],"transformation":[46],"Burgers":[49],"equation.":[50]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
