{"id":"https://openalex.org/W2167606115","doi":"https://doi.org/10.1137/140960803","title":"``Entropic\u201d Solutions to a Thermodynamically Consistent PDE System for Phase Transitions and Damage","display_name":"``Entropic\u201d Solutions to a Thermodynamically Consistent PDE System for Phase Transitions and Damage","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2167606115","doi":"https://doi.org/10.1137/140960803","mag":"2167606115"},"language":"en","primary_location":{"id":"doi:10.1137/140960803","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140960803","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":true,"oa_status":"green","oa_url":"http://hdl.handle.net/2434/345085","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5017446516","display_name":"Elisabetta Rocca","orcid":"https://orcid.org/0000-0002-9930-907X"},"institutions":[{"id":"https://openalex.org/I189158943","display_name":"University of Milan","ror":"https://ror.org/00wjc7c48","country_code":"IT","type":"education","lineage":["https://openalex.org/I189158943"]},{"id":"https://openalex.org/I4210092021","display_name":"Weierstrass Institute for Applied Analysis and Stochastics","ror":"https://ror.org/00h1x4t21","country_code":"DE","type":"facility","lineage":["https://openalex.org/I315704651","https://openalex.org/I4210092021"]}],"countries":["DE","IT"],"is_corresponding":true,"raw_author_name":"Elisabetta Rocca","raw_affiliation_strings":["Dipartimento di Matematica, Universit\u00e0 di Milano, I-20133, Milan, Italy","Weierstrass Institute for Applied Analysis and Stochastics, D-10117, Berlin, Germany, and"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica, Universit\u00e0 di Milano, I-20133, Milan, Italy","institution_ids":["https://openalex.org/I189158943"]},{"raw_affiliation_string":"Weierstrass Institute for Applied Analysis and Stochastics, D-10117, Berlin, Germany, and","institution_ids":["https://openalex.org/I4210092021"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5031245511","display_name":"Riccarda Rossi","orcid":"https://orcid.org/0000-0002-7808-0261"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Riccarda Rossi","raw_affiliation_strings":["DICATAM-Sezione di Matematica, Universit\u00e0 di"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"DICATAM-Sezione di Matematica, Universit\u00e0 di","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5017446516"],"corresponding_institution_ids":["https://openalex.org/I189158943","https://openalex.org/I4210092021"],"apc_list":null,"apc_paid":null,"fwci":6.8199,"has_fulltext":false,"cited_by_count":39,"citation_normalized_percentile":{"value":0.96643972,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"47","issue":"4","first_page":"2519","last_page":"2586"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9994999766349792,"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"}},{"id":"https://openalex.org/T11366","display_name":"Elasticity and Material Modeling","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"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/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.9958000183105469,"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/mathematics","display_name":"Mathematics","score":0.8002853393554688},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.7272835373878479},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6639609336853027},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.6604626774787903},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.5416046380996704},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.49400120973587036},{"id":"https://openalex.org/keywords/phase-transition","display_name":"Phase transition","score":0.4889925718307495},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.4375167489051819},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.41226714849472046},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.3305395841598511},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.19946905970573425},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08512607216835022},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07938486337661743}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8002853393554688},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.7272835373878479},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6639609336853027},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.6604626774787903},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.5416046380996704},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.49400120973587036},{"id":"https://openalex.org/C149288129","wikidata":"https://www.wikidata.org/wiki/Q185357","display_name":"Phase transition","level":2,"score":0.4889925718307495},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.4375167489051819},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.41226714849472046},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3305395841598511},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.19946905970573425},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08512607216835022},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07938486337661743},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/140960803","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140960803","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:air.unimi.it:2434/345085","is_oa":true,"landing_page_url":"http://hdl.handle.net/2434/345085","pdf_url":null,"source":{"id":"https://openalex.org/S4306400516","display_name":"Archivio Istituzionale della Ricerca (Universita Degli Studi Di Milano)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I189158943","host_organization_name":"University of Milan","host_organization_lineage":["https://openalex.org/I189158943"],"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"},{"id":"pmh:oai:iris.unibs.it:11379/463186","is_oa":false,"landing_page_url":"http://hdl.handle.net/11379/463186","pdf_url":null,"source":{"id":"https://openalex.org/S4306400804","display_name":"Institutional Research Information System (Universit\u00e0 degli Studi di Brescia)","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:air.unimi.it:2434/345085","is_oa":true,"landing_page_url":"http://hdl.handle.net/2434/345085","pdf_url":null,"source":{"id":"https://openalex.org/S4306400516","display_name":"Archivio Istituzionale della Ricerca (Universita Degli Studi Di Milano)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I189158943","host_organization_name":"University of Milan","host_organization_lineage":["https://openalex.org/I189158943"],"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":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.6800000071525574}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":51,"referenced_works":["https://openalex.org/W51359577","https://openalex.org/W1515685921","https://openalex.org/W1525219438","https://openalex.org/W1582617215","https://openalex.org/W1591473593","https://openalex.org/W1597209545","https://openalex.org/W1604809219","https://openalex.org/W1801863981","https://openalex.org/W1967206000","https://openalex.org/W1969464363","https://openalex.org/W1973108246","https://openalex.org/W1973551984","https://openalex.org/W1985501684","https://openalex.org/W1987385435","https://openalex.org/W1989046305","https://openalex.org/W1996353338","https://openalex.org/W1996567122","https://openalex.org/W1998015501","https://openalex.org/W2001280426","https://openalex.org/W2001600827","https://openalex.org/W2006129860","https://openalex.org/W2017132259","https://openalex.org/W2024403296","https://openalex.org/W2031975369","https://openalex.org/W2044414647","https://openalex.org/W2044637889","https://openalex.org/W2057410521","https://openalex.org/W2057932257","https://openalex.org/W2068432368","https://openalex.org/W2071014770","https://openalex.org/W2073078790","https://openalex.org/W2081762789","https://openalex.org/W2081919047","https://openalex.org/W2099873196","https://openalex.org/W2118031691","https://openalex.org/W2118772813","https://openalex.org/W2119951320","https://openalex.org/W2124471636","https://openalex.org/W2132228518","https://openalex.org/W2137654124","https://openalex.org/W2142690570","https://openalex.org/W2147753909","https://openalex.org/W2153024959","https://openalex.org/W2161867970","https://openalex.org/W2165902635","https://openalex.org/W2963743170","https://openalex.org/W2963806863","https://openalex.org/W2964236671","https://openalex.org/W3103670900","https://openalex.org/W3106294346","https://openalex.org/W4256110609"],"related_works":["https://openalex.org/W2006251942","https://openalex.org/W2364741597","https://openalex.org/W1492103595","https://openalex.org/W1864774435","https://openalex.org/W946352265","https://openalex.org/W3020787026","https://openalex.org/W2334479858","https://openalex.org/W2799209613","https://openalex.org/W1971388572","https://openalex.org/W1507702947"],"abstract_inverted_index":{"In":[0],"this":[1,176],"paper":[2],"we":[3],"analyze":[4],"a":[5,55,64,88,139,159],"PDE":[6,150],"system":[7,53,151],"modeling":[8],"(nonisothermal)":[9],"phase":[10],"transitions":[11],"and":[12,86,156],"damage":[13],"phenomena":[14],"in":[15,23,33,49,138],"thermoviscoelastic":[16],"materials.":[17],"The":[18,51],"model":[19],"is":[20,29,77],"thermodynamically":[21],"consistent:":[22],"particular,":[24],"no":[25],"small":[26],"perturbation":[27],"assumption":[28],"adopted,":[30],"which":[31],"results":[32,132],"the":[34,40,44,61,74,80,96,104,108,123,126,136,145,149,158,172],"presence":[35],"of":[36,43,63,67,82,87,99,107,125,148,157,175],"quadratic":[37],"terms":[38],"on":[39,122,163],"right-hand":[41],"side":[42],"temperature":[45,75],"equation,":[46],"only":[47],"estimated":[48],"$L^1$.":[50],"whole":[52],"has":[54],"highly":[56],"nonlinear":[57,146],"character.":[58],"We":[59,129],"address":[60],"existence":[62,116],"weak":[65],"notion":[66],"solution,":[68],"referred":[69],"to":[70,113,135,144],"as":[71,101,103],"\u201centropic,\u201d":[72],"where":[73],"equation":[76],"formulated":[78],"with":[79],"aid":[81],"an":[83],"entropy":[84],"inequality,":[85],"total":[89],"energy":[90],"inequality.":[91],"This":[92],"solvability":[93],"concept":[94],"reflects":[95],"basic":[97],"principles":[98],"thermomechanics,":[100],"well":[102],"thermodynamical":[105],"consistency":[106],"model.":[109,177],"It":[110],"allows":[111],"us":[112],"obtain":[114],"global-in-time":[115],"theorems":[117],"without":[118],"imposing":[119],"any":[120],"restriction":[121],"size":[124],"initial":[127],"data.":[128],"prove":[130],"our":[131],"by":[133],"passing":[134],"limit":[137],"time-discretization":[140],"scheme,":[141],"carefully":[142],"tailored":[143],"features":[147],"(with":[152],"its":[153],"entropic":[154],"formulation)":[155],"priori":[160],"estimates":[161],"performed":[162],"it.":[164],"Our":[165],"time-discrete":[166],"analysis":[167],"could":[168],"be":[169],"useful":[170],"toward":[171],"numerical":[173],"study":[174]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":9},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":2}],"updated_date":"2026-08-11T07:18:39.950985","created_date":"2025-10-10T00:00:00"}