{"id":"https://openalex.org/W2997691404","doi":"https://doi.org/10.1137/19m1266836","title":"On the \u201cVacuum\u201d Dam-Break Problem: Exact Solutions and Their Long Time Asymptotics","display_name":"On the \u201cVacuum\u201d Dam-Break Problem: Exact Solutions and Their Long Time Asymptotics","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W2997691404","doi":"https://doi.org/10.1137/19m1266836","mag":"2997691404"},"language":"en","primary_location":{"id":"doi:10.1137/19m1266836","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1266836","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://hdl.handle.net/10281/263065","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051195478","display_name":"Roberto Camassa","orcid":"https://orcid.org/0000-0002-1607-5950"},"institutions":[{"id":"https://openalex.org/I11039511","display_name":"University of Bergamo","ror":"https://ror.org/02mbd5571","country_code":"IT","type":"education","lineage":["https://openalex.org/I11039511"]},{"id":"https://openalex.org/I114027177","display_name":"University of North Carolina at Chapel Hill","ror":"https://ror.org/0130frc33","country_code":"US","type":"education","lineage":["https://openalex.org/I114027177"]},{"id":"https://openalex.org/I66752286","display_name":"University of Milano-Bicocca","ror":"https://ror.org/01ynf4891","country_code":"IT","type":"education","lineage":["https://openalex.org/I66752286"]}],"countries":["IT","US"],"is_corresponding":true,"raw_author_name":"Roberto Camassa","raw_affiliation_strings":["\\S Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Universita' di Bergamo-Dalmine, Dalmine (BG), Italy 24044","\\ddagger Dipartimento di Matematica e Applicazioni, Universita' di Milano-Bicocca, Milano, Italy 20125","\\dagger Department of Mathematics, Carolina Center for Interdisciplinary Applied Mathematics Univer-sity of North Carolina at Chapel Hill, Chapel Hill, North Carolina, 27599"],"raw_orcid":"https://orcid.org/0000-0002-1607-5950","affiliations":[{"raw_affiliation_string":"\\S Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Universita' di Bergamo-Dalmine, Dalmine (BG), Italy 24044","institution_ids":["https://openalex.org/I11039511"]},{"raw_affiliation_string":"\\ddagger Dipartimento di Matematica e Applicazioni, Universita' di Milano-Bicocca, Milano, Italy 20125","institution_ids":["https://openalex.org/I66752286"]},{"raw_affiliation_string":"\\dagger Department of Mathematics, Carolina Center for Interdisciplinary Applied Mathematics Univer-sity of North Carolina at Chapel Hill, Chapel Hill, North Carolina, 27599","institution_ids":["https://openalex.org/I114027177"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031219622","display_name":"Gregorio Falqui","orcid":"https://orcid.org/0000-0002-4893-9186"},"institutions":[{"id":"https://openalex.org/I11039511","display_name":"University of Bergamo","ror":"https://ror.org/02mbd5571","country_code":"IT","type":"education","lineage":["https://openalex.org/I11039511"]},{"id":"https://openalex.org/I114027177","display_name":"University of North Carolina at Chapel Hill","ror":"https://ror.org/0130frc33","country_code":"US","type":"education","lineage":["https://openalex.org/I114027177"]},{"id":"https://openalex.org/I66752286","display_name":"University of Milano-Bicocca","ror":"https://ror.org/01ynf4891","country_code":"IT","type":"education","lineage":["https://openalex.org/I66752286"]}],"countries":["IT","US"],"is_corresponding":false,"raw_author_name":"Gregorio Falqui","raw_affiliation_strings":["\\dagger Department of Mathematics, Carolina Center for Interdisciplinary Applied Mathematics Univer-sity of North Carolina at Chapel Hill, Chapel Hill, North Carolina, 27599","\\ddagger Dipartimento di Matematica e Applicazioni, Universita' di Milano-Bicocca, Milano, Italy 20125","\\S Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Universita' di Bergamo-Dalmine, Dalmine (BG), Italy 24044"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"\\dagger Department of Mathematics, Carolina Center for Interdisciplinary Applied Mathematics Univer-sity of North Carolina at Chapel Hill, Chapel Hill, North Carolina, 27599","institution_ids":["https://openalex.org/I114027177"]},{"raw_affiliation_string":"\\ddagger Dipartimento di Matematica e Applicazioni, Universita' di Milano-Bicocca, Milano, Italy 20125","institution_ids":["https://openalex.org/I66752286"]},{"raw_affiliation_string":"\\S Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Universita' di Bergamo-Dalmine, Dalmine (BG), Italy 24044","institution_ids":["https://openalex.org/I11039511"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5043732061","display_name":"G. Ortenzi","orcid":"https://orcid.org/0000-0003-2192-6737"},"institutions":[{"id":"https://openalex.org/I11039511","display_name":"University of Bergamo","ror":"https://ror.org/02mbd5571","country_code":"IT","type":"education","lineage":["https://openalex.org/I11039511"]},{"id":"https://openalex.org/I114027177","display_name":"University of North Carolina at Chapel Hill","ror":"https://ror.org/0130frc33","country_code":"US","type":"education","lineage":["https://openalex.org/I114027177"]},{"id":"https://openalex.org/I66752286","display_name":"University of Milano-Bicocca","ror":"https://ror.org/01ynf4891","country_code":"IT","type":"education","lineage":["https://openalex.org/I66752286"]}],"countries":["IT","US"],"is_corresponding":false,"raw_author_name":"Giovanni Ortenzi","raw_affiliation_strings":["\\S Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Universita' di Bergamo-Dalmine, Dalmine (BG), Italy 24044","\\dagger Department of Mathematics, Carolina Center for Interdisciplinary Applied Mathematics Univer-sity of North Carolina at Chapel Hill, Chapel Hill, North Carolina, 27599","\\ddagger Dipartimento di Matematica e Applicazioni, Universita' di Milano-Bicocca, Milano, Italy 20125"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"\\S Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Universita' di Bergamo-Dalmine, Dalmine (BG), Italy 24044","institution_ids":["https://openalex.org/I11039511"]},{"raw_affiliation_string":"\\dagger Department of Mathematics, Carolina Center for Interdisciplinary Applied Mathematics Univer-sity of North Carolina at Chapel Hill, Chapel Hill, North Carolina, 27599","institution_ids":["https://openalex.org/I114027177"]},{"raw_affiliation_string":"\\ddagger Dipartimento di Matematica e Applicazioni, Universita' di Milano-Bicocca, Milano, Italy 20125","institution_ids":["https://openalex.org/I66752286"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020573834","display_name":"Marco Pedroni","orcid":"https://orcid.org/0000-0002-7358-0945"},"institutions":[{"id":"https://openalex.org/I11039511","display_name":"University of Bergamo","ror":"https://ror.org/02mbd5571","country_code":"IT","type":"education","lineage":["https://openalex.org/I11039511"]},{"id":"https://openalex.org/I114027177","display_name":"University of North Carolina at Chapel Hill","ror":"https://ror.org/0130frc33","country_code":"US","type":"education","lineage":["https://openalex.org/I114027177"]},{"id":"https://openalex.org/I66752286","display_name":"University of Milano-Bicocca","ror":"https://ror.org/01ynf4891","country_code":"IT","type":"education","lineage":["https://openalex.org/I66752286"]}],"countries":["IT","US"],"is_corresponding":false,"raw_author_name":"Marco Pedroni","raw_affiliation_strings":["\\ddagger Dipartimento di Matematica e Applicazioni, Universita' di Milano-Bicocca, Milano, Italy 20125","\\dagger Department of Mathematics, Carolina Center for Interdisciplinary Applied Mathematics Univer-sity of North Carolina at Chapel Hill, Chapel Hill, North Carolina, 27599","\\S Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Universita' di Bergamo-Dalmine, Dalmine (BG), Italy 24044"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"\\ddagger Dipartimento di Matematica e Applicazioni, Universita' di Milano-Bicocca, Milano, Italy 20125","institution_ids":["https://openalex.org/I66752286"]},{"raw_affiliation_string":"\\dagger Department of Mathematics, Carolina Center for Interdisciplinary Applied Mathematics Univer-sity of North Carolina at Chapel Hill, Chapel Hill, North Carolina, 27599","institution_ids":["https://openalex.org/I114027177"]},{"raw_affiliation_string":"\\S Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Universita' di Bergamo-Dalmine, Dalmine (BG), Italy 24044","institution_ids":["https://openalex.org/I11039511"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5005684785","display_name":"Giuseppe Pitton","orcid":"https://orcid.org/0000-0002-2245-8587"},"institutions":[{"id":"https://openalex.org/I47508984","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10","country_code":"GB","type":"education","lineage":["https://openalex.org/I47508984"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Giuseppe Pitton","raw_affiliation_strings":["Department of Mathematics, Imperial College, London, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Imperial College, London, UK","institution_ids":["https://openalex.org/I47508984"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5051195478"],"corresponding_institution_ids":["https://openalex.org/I11039511","https://openalex.org/I114027177","https://openalex.org/I66752286"],"apc_list":null,"apc_paid":null,"fwci":0.3594,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.52303284,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"80","issue":"1","first_page":"44","last_page":"70"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.9945999979972839,"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"}},"topics":[{"id":"https://openalex.org/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.9945999979972839,"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/T10940","display_name":"Navier-Stokes equation solutions","score":0.9926999807357788,"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":0.9886999726295471,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/heaviside-step-function","display_name":"Heaviside step function","score":0.7997696399688721},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6826784610748291},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.584416389465332},{"id":"https://openalex.org/keywords/jump","display_name":"Jump","score":0.5778297185897827},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.5522206425666809},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.5179561376571655},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.45052486658096313},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.44914180040359497},{"id":"https://openalex.org/keywords/gravitational-singularity","display_name":"Gravitational singularity","score":0.42934805154800415},{"id":"https://openalex.org/keywords/rarefaction","display_name":"Rarefaction (ecology)","score":0.41452592611312866},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2207278609275818},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.19442591071128845}],"concepts":[{"id":"https://openalex.org/C7980502","wikidata":"https://www.wikidata.org/wiki/Q322339","display_name":"Heaviside step function","level":2,"score":0.7997696399688721},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6826784610748291},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.584416389465332},{"id":"https://openalex.org/C2780695682","wikidata":"https://www.wikidata.org/wiki/Q4005959","display_name":"Jump","level":2,"score":0.5778297185897827},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.5522206425666809},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.5179561376571655},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.45052486658096313},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.44914180040359497},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.42934805154800415},{"id":"https://openalex.org/C132705693","wikidata":"https://www.wikidata.org/wiki/Q7294543","display_name":"Rarefaction (ecology)","level":3,"score":0.41452592611312866},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2207278609275818},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.19442591071128845},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"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/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"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/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/19m1266836","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1266836","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:aisberg.unibg.it:10446/150156","is_oa":false,"landing_page_url":"http://hdl.handle.net/10446/150156","pdf_url":null,"source":{"id":"https://openalex.org/S4377196347","display_name":"Aisberg (University of Bergamo)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I11039511","host_organization_name":"University of Bergamo","host_organization_lineage":["https://openalex.org/I11039511"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:boa.unimib.it:10281/263065","is_oa":true,"landing_page_url":"https://hdl.handle.net/10281/263065","pdf_url":null,"source":{"id":"https://openalex.org/S4306401259","display_name":"BOA (University of Milano-Bicocca)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I66752286","host_organization_name":"University of Milano-Bicocca","host_organization_lineage":["https://openalex.org/I66752286"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:boa.unimib.it:10281/263065","is_oa":true,"landing_page_url":"https://hdl.handle.net/10281/263065","pdf_url":null,"source":{"id":"https://openalex.org/S4306401259","display_name":"BOA (University of Milano-Bicocca)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I66752286","host_organization_name":"University of Milano-Bicocca","host_organization_lineage":["https://openalex.org/I66752286"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2782603331","display_name":null,"funder_award_id":"682603","funder_id":"https://openalex.org/F4320338335","funder_display_name":"H2020 European Research Council"},{"id":"https://openalex.org/G3120699348","display_name":null,"funder_award_id":"RTG DMS-0943851","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4025299300","display_name":null,"funder_award_id":"DMS-1009750","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5526348500","display_name":null,"funder_award_id":"N00014-18-1-12490","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G5990409462","display_name":null,"funder_award_id":"N00014-12-0749","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G8074822591","display_name":null,"funder_award_id":"CMG ARC-1025523","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8174154610","display_name":null,"funder_award_id":"DMS - 1517879","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8977015700","display_name":null,"funder_award_id":"778010","funder_id":"https://openalex.org/F4320338335","funder_display_name":"H2020 European Research Council"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"},{"id":"https://openalex.org/F4320337345","display_name":"Office of Naval Research","ror":"https://ror.org/00rk2pe57"},{"id":"https://openalex.org/F4320338335","display_name":"H2020 European Research Council","ror":"https://ror.org/0472cxd90"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1970228368","https://openalex.org/W1983195644","https://openalex.org/W1985107412","https://openalex.org/W1992913018","https://openalex.org/W1999987757","https://openalex.org/W2004954372","https://openalex.org/W2019680912","https://openalex.org/W2053952260","https://openalex.org/W2078290517","https://openalex.org/W2080049628","https://openalex.org/W2080705460","https://openalex.org/W2089139620","https://openalex.org/W2094553410","https://openalex.org/W2116729868","https://openalex.org/W2288772602","https://openalex.org/W2903759772","https://openalex.org/W2953995959","https://openalex.org/W2963366101","https://openalex.org/W3104259856"],"related_works":["https://openalex.org/W2074084660","https://openalex.org/W2901390526","https://openalex.org/W2088409529","https://openalex.org/W2077007836","https://openalex.org/W4253354271","https://openalex.org/W2018118488","https://openalex.org/W4390601309","https://openalex.org/W2333500828","https://openalex.org/W1922048481","https://openalex.org/W1594995899"],"abstract_inverted_index":{"The":[0],"classical":[1],"dam-break":[2],"problem":[3],"for":[4,103,121,179],"the":[5,17,24,50,54,58,65,68,79,91,104,107,110,131,146,157,180,191,210,219,222],"shallow":[6],"water":[7],"system":[8],"with":[9,90,113,209],"a":[10,28,97,118,126,140,163,226],"dry/vacuum":[11],"downstream":[12],"state":[13,112],"is":[14,72,206],"revisited":[15],"in":[16,87,124,174,186,188],"context":[18],"of":[19,27,49,212,221],"exact":[20],"solutions":[21],"which":[22],"generalize":[23],"Riemann":[25],"setup":[26],"Heaviside":[29],"jump":[30],"between":[31],"constant":[32],"states":[33],"to":[34,44,149,190,217,225],"continuous":[35,74],"initial":[36],"data.":[37],"Two":[38],"main":[39],"setups":[40],"are":[41,184,198],"considered,":[42],"chosen":[43],"illustrate":[45],"how":[46],"local":[47],"properties":[48],"dependent":[51],"variables":[52,216],"at":[53,78,130,156,162,170,200],"vacuum":[55],"point":[56,134,148],"influence":[57],"evolution":[59,93,205],"over":[60],"different":[61],"time":[62,92,125,204],"scales.":[63],"For":[64],"first":[66,115],"case,":[67,106,193],"elevation":[69,108],"(density)":[70],"variable":[71],"initially":[73],"but":[75],"not":[76],"differentiable":[77],"dry":[80,111,158],"(vacuum)":[81],"point:":[82],"no":[83],"gradient":[84,128],"catastrophe":[85,129],"develops":[86],"this":[88],"variable,":[89],"eventually":[94],"merging":[95],"into":[96,151],"single":[98],"(shifted)":[99],"Stoker":[100,141],"wave.":[101,229],"Conversely,":[102],"second":[105],"joins":[109],"vanishing":[114],"derivative":[116],"and":[117,136,160],"curvature":[119,167],"jump:":[120],"an":[122],"instant":[123],"global":[127],"fixed":[132,164],"contact":[133,147],"forms":[135],"immediately":[137],"evolves":[138],"as":[139],"parabolic":[142],"simple":[143],"wave,":[144],"allowing":[145],"split":[150],"two":[152],"moving":[153],"points,":[154],"one":[155,161],"bed":[159],"elevation,":[165],"where":[166],"singularities":[168],"persist":[169],"all":[171,201],"times.":[172,202],"Although":[173],"both":[175],"cases":[176],"shocks":[177],"develop":[178],"velocity":[181],"field,":[182],"these":[183],"nongeneric,":[185],"that,":[187],"contrast":[189],"usual":[192],"infinitely":[194],"many":[195],"conservation":[196],"laws":[197],"satisfied":[199],"Long":[203],"further":[207],"analyzed":[208],"help":[211],"new":[213],"stretched":[214],"\u201cunfolding\u201d":[215],"extract":[218],"details":[220],"asymptotic":[223],"approach":[224],"rarefaction":[227],"Stoker-like":[228]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
