{"id":"https://openalex.org/W1971534614","doi":"https://doi.org/10.1112/jlms/jdr020","title":"Gradient estimates for degenerate quasi-linear parabolic equations","display_name":"Gradient estimates for degenerate quasi-linear parabolic equations","publication_year":2011,"publication_date":"2011-08-19","ids":{"openalex":"https://openalex.org/W1971534614","doi":"https://doi.org/10.1112/jlms/jdr020","mag":"1971534614"},"language":"en","primary_location":{"id":"doi:10.1112/jlms/jdr020","is_oa":false,"landing_page_url":"https://doi.org/10.1112/jlms/jdr020","pdf_url":null,"source":{"id":"https://openalex.org/S4210207695","display_name":"Journal of the London Mathematical Society","issn_l":"0024-6107","issn":["0024-6107","1469-7750"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of the London Mathematical Society","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1004.3046","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5083007415","display_name":"Vitali Liskevich","orcid":null},"institutions":[{"id":"https://openalex.org/I39586589","display_name":"Swansea University","ror":"https://ror.org/053fq8t95","country_code":"GB","type":"education","lineage":["https://openalex.org/I39586589"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Vitali Liskevich","raw_affiliation_strings":["Department of Mathematics, Swansea University, Swansea, SA2 8PP, United Kingdom","Department of MathematicsSwansea UniversitySwanseaSA2 8PPUnited Kingdom"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Swansea University, Swansea, SA2 8PP, United Kingdom","institution_ids":["https://openalex.org/I39586589"]},{"raw_affiliation_string":"Department of MathematicsSwansea UniversitySwanseaSA2 8PPUnited Kingdom","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5022312643","display_name":"Igor I. Skrypnik","orcid":"https://orcid.org/0000-0002-7398-9273"},"institutions":[{"id":"https://openalex.org/I4210089221","display_name":"Institute of Applied Mathematics and Mechanics","ror":"https://ror.org/006c77f48","country_code":"UA","type":"facility","lineage":["https://openalex.org/I149851306","https://openalex.org/I4210089221"]}],"countries":["UA"],"is_corresponding":false,"raw_author_name":"Igor I. Skrypnik","raw_affiliation_strings":["Institute of Applied Mathematics and Mechanics, Donetsk 83114, Ukraine, [email protected].donbass.com","Institute of Applied Mathematics and MechanicsDonetsk 83114Ukraine"],"affiliations":[{"raw_affiliation_string":"Institute of Applied Mathematics and Mechanics, Donetsk 83114, Ukraine, [email protected].donbass.com","institution_ids":["https://openalex.org/I4210089221"]},{"raw_affiliation_string":"Institute of Applied Mathematics and MechanicsDonetsk 83114Ukraine","institution_ids":["https://openalex.org/I4210089221"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5050932795","display_name":"Zeev Sobol","orcid":"https://orcid.org/0000-0003-4862-427X"},"institutions":[{"id":"https://openalex.org/I39586589","display_name":"Swansea University","ror":"https://ror.org/053fq8t95","country_code":"GB","type":"education","lineage":["https://openalex.org/I39586589"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Zeev Sobol","raw_affiliation_strings":["Department of Mathematics, Swansea University, Swansea, SA2 8PP, United Kingdom, [email protected]","Department of MathematicsSwansea UniversitySwanseaSA2 8PPUnited Kingdom"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Swansea University, Swansea, SA2 8PP, United Kingdom, [email protected]","institution_ids":["https://openalex.org/I39586589"]},{"raw_affiliation_string":"Department of MathematicsSwansea UniversitySwanseaSA2 8PPUnited Kingdom","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5083007415"],"corresponding_institution_ids":["https://openalex.org/I39586589"],"apc_list":{"value":3710,"currency":"USD","value_usd":3710},"apc_paid":null,"fwci":1.2322,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.74781818,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"84","issue":"2","first_page":"446","last_page":"474"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9994999766349792,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9994999766349792,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.9937000274658203,"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/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.8156715035438538},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7777309417724609},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.7653415203094482},{"id":"https://openalex.org/keywords/divergence","display_name":"Divergence (linguistics)","score":0.7440483570098877},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.6552687883377075},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.6200892925262451},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.5680978298187256},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.559156060218811},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.46608230471611023},{"id":"https://openalex.org/keywords/parabolic-partial-differential-equation","display_name":"Parabolic partial differential equation","score":0.4606464207172394},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.42829224467277527},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4180294871330261},{"id":"https://openalex.org/keywords/laplace-operator","display_name":"Laplace operator","score":0.4135439991950989},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1562156081199646},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.12733805179595947}],"concepts":[{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.8156715035438538},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7777309417724609},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.7653415203094482},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.7440483570098877},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6552687883377075},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.6200892925262451},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.5680978298187256},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.559156060218811},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.46608230471611023},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.4606464207172394},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.42829224467277527},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4180294871330261},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.4135439991950989},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1562156081199646},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.12733805179595947},{"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/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","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/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1112/jlms/jdr020","is_oa":false,"landing_page_url":"https://doi.org/10.1112/jlms/jdr020","pdf_url":null,"source":{"id":"https://openalex.org/S4210207695","display_name":"Journal of the London Mathematical Society","issn_l":"0024-6107","issn":["0024-6107","1469-7750"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of the London Mathematical Society","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1004.3046","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1004.3046","pdf_url":"https://arxiv.org/pdf/1004.3046","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"pmh:oai:cronfa.swan.ac.uk:cronfa7700","is_oa":false,"landing_page_url":"https://cronfa.swan.ac.uk/Record/cronfa7700","pdf_url":null,"source":{"id":"https://openalex.org/S4306401612","display_name":"Cronfa (Swansea University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I39586589","host_organization_name":"Swansea University","host_organization_lineage":["https://openalex.org/I39586589"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1004.3046","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1004.3046","pdf_url":"https://arxiv.org/pdf/1004.3046","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W65402155","https://openalex.org/W97097846","https://openalex.org/W185753331","https://openalex.org/W593307463","https://openalex.org/W644710670","https://openalex.org/W1480253648","https://openalex.org/W1504810697","https://openalex.org/W1515091993","https://openalex.org/W1609275808","https://openalex.org/W1977355332","https://openalex.org/W1984815757","https://openalex.org/W1990902226","https://openalex.org/W1996578703","https://openalex.org/W2012063459","https://openalex.org/W2021164513","https://openalex.org/W2025854194","https://openalex.org/W2030521634","https://openalex.org/W2035163028","https://openalex.org/W2063447685","https://openalex.org/W2065546409","https://openalex.org/W2072685235","https://openalex.org/W2073078790","https://openalex.org/W2369386221","https://openalex.org/W2495272354","https://openalex.org/W2577909137","https://openalex.org/W2950031618","https://openalex.org/W2964346640","https://openalex.org/W4231119765","https://openalex.org/W4285719527","https://openalex.org/W4299835338"],"related_works":["https://openalex.org/W1677518548","https://openalex.org/W4286857066","https://openalex.org/W3213958552","https://openalex.org/W4238239967","https://openalex.org/W4297812081","https://openalex.org/W2949926331","https://openalex.org/W2754545762","https://openalex.org/W4299562540","https://openalex.org/W4298208753","https://openalex.org/W3175273680"],"abstract_inverted_index":{"For":[0],"a":[1],"class":[2],"of":[3,29],"divergence-type":[4],"quasi-linear":[5],"degenerate":[6],"parabolic":[7],"equations":[8],"with":[9,18,48],"differentiable":[10],"structure":[11],"and":[12,31],"lower":[13,34],"order":[14,35],"coefficients":[15,36],"form":[16],"small":[17],"respect":[19,49],"to":[20,50],"the":[21,27,33,43,51],"Laplacian":[22],"we":[23,40],"obtain":[24],"Lq-estimates":[25],"for":[26,32],"gradients":[28],"solutions,":[30],"from":[37],"Kato-type":[38],"classes":[39],"show":[41],"that":[42],"solutions":[44],"are":[45],"Lipschitz":[46],"continuous":[47],"space":[52],"variable.":[53]},"counts_by_year":[{"year":2021,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
