{"id":"https://openalex.org/W7221148136","doi":"https://doi.org/10.48550/arxiv.2610.11462","title":"Global existence of weak solutions in a Keller-Segel system with local anisotropy and nonlinear diffusion","display_name":"Global existence of weak solutions in a Keller-Segel system with local anisotropy and nonlinear diffusion","publication_year":2026,"publication_date":"2026-10-08","ids":{"openalex":"https://openalex.org/W7221148136","doi":"https://doi.org/10.48550/arxiv.2610.11462"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.11462","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.11462","pdf_url":"https://arxiv.org/pdf/2610.11462","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/2610.11462","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5158037443","display_name":"Max Hegenbart","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I114112103","display_name":"Leibniz University Hannover","ror":"https://ror.org/0304hq317","country_code":"DE","type":"education","lineage":["https://openalex.org/I114112103"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Hegenbart, Max","raw_affiliation_strings":["Leibniz Universit\u00e4t Hannover Institut f\u00fcr Angewandte Mathematik"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Leibniz Universit\u00e4t Hannover Institut f\u00fcr Angewandte Mathematik","institution_ids":["https://openalex.org/I114112103"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5158037443"],"corresponding_institution_ids":["https://openalex.org/I114112103"],"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/T11829","display_name":"Mathematical Biology Tumor Growth","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T11829","display_name":"Mathematical Biology Tumor Growth","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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":0.0005000000237487257,"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/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.00019999999494757503,"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/keller-segel-model","display_name":"Keller-Segel model","score":0.9959999918937683},{"id":"https://openalex.org/keywords/global-weak-solutions","display_name":"global weak solutions","score":0.7950000166893005},{"id":"https://openalex.org/keywords/nonlinear-diffusion","display_name":"nonlinear diffusion","score":0.984000027179718},{"id":"https://openalex.org/keywords/weak-convergence","display_name":"weak convergence","score":0.5809999704360962},{"id":"https://openalex.org/keywords/neumann-boundary-conditions","display_name":"Neumann boundary conditions","score":0.7089999914169312},{"id":"https://openalex.org/keywords/chemotaxis","display_name":"chemotaxis","score":0.6740000247955322},{"id":"https://openalex.org/keywords/bounded-domains","display_name":"bounded domains","score":0.49900001287460327},{"id":"https://openalex.org/keywords/weak-solutions","display_name":"weak solutions","score":0.3269999921321869},{"id":"https://openalex.org/keywords/l-infinity-estimates","display_name":"L-infinity estimates","score":0.4869999885559082},{"id":"https://openalex.org/keywords/weak-convergence-of-measures","display_name":"weak convergence of measures","score":0.5210000276565552}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7677000164985657},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.7512000203132629},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6812999844551086},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6097000241279602},{"id":"https://openalex.org/C45137528","wikidata":"https://www.wikidata.org/wiki/Q7977966","display_name":"Weak solution","level":2,"score":0.5993000268936157},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5482000112533569},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5473999977111816},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.5440999865531921},{"id":"https://openalex.org/C57945734","wikidata":"https://www.wikidata.org/wiki/Q7977941","display_name":"Weak convergence","level":3,"score":0.4966000020503998},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.4810999929904938},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4442000091075897},{"id":"https://openalex.org/C163681178","wikidata":"https://www.wikidata.org/wiki/Q1149279","display_name":"Neumann boundary condition","level":3,"score":0.42750000953674316},{"id":"https://openalex.org/C85725439","wikidata":"https://www.wikidata.org/wiki/Q466686","display_name":"Anisotropy","level":2,"score":0.40849998593330383},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.4041999876499176},{"id":"https://openalex.org/C203504353","wikidata":"https://www.wikidata.org/wiki/Q4765461","display_name":"Anisotropic diffusion","level":3,"score":0.39410001039505005},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.3797000050544739},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.3531999886035919},{"id":"https://openalex.org/C153919410","wikidata":"https://www.wikidata.org/wiki/Q9993851","display_name":"Weak formulation","level":3,"score":0.3321000039577484},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3028999865055084},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.2921999990940094},{"id":"https://openalex.org/C121231716","wikidata":"https://www.wikidata.org/wiki/Q901210","display_name":"Reaction\u2013diffusion system","level":2,"score":0.2775000035762787},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.26499998569488525},{"id":"https://openalex.org/C80469333","wikidata":"https://www.wikidata.org/wiki/Q189088","display_name":"Von Neumann architecture","level":2,"score":0.257999986410141}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.11462","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.11462","pdf_url":"https://arxiv.org/pdf/2610.11462","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.2610.11462","doi":"https://doi.org/10.48550/arxiv.2610.11462","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.11462","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:2610.11462","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.11462","pdf_url":"https://arxiv.org/pdf/2610.11462","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":[],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7221148136.pdf","grobid_xml":"https://content.openalex.org/works/W7221148136.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W4407669626","https://openalex.org/W7163718742"],"related_works":["https://openalex.org/W7163718742","https://openalex.org/W4296554882","https://openalex.org/W4384207783","https://openalex.org/W7163983967","https://openalex.org/W7202258862","https://openalex.org/W7081954974","https://openalex.org/W2802186118","https://openalex.org/W2607374977","https://openalex.org/W4388197415","https://openalex.org/W4412552798"],"abstract_inverted_index":{"We":[0],"show":[1,54,69],"global":[2,70],"existence":[3,71],"of":[4,11,56,74,106,109,115,120,128,131],"weak":[5,107,113,129],"solutions":[6,76,108,114],"to":[7,68,78,98,112,135],"the":[8,79,99,110,116,121,126],"following":[9],"system":[10,81],"partial":[12],"differential":[13],"equations":[14,20],"with":[15,43],"nonlinear":[16],"diffusion":[17],"in":[18,36,58,62,125,138],"both":[19],"and":[21,47,60,72],"anisotropic":[22],"signal":[23],"production":[24],"\\begin{equation}":[25,82],"\\begin{cases}":[26,83],"u_t=\\nabla\\cdot((u+1)^{m_1-1}\\nabla":[27],"u)-\\nabla\\cdot(u\\nabla":[28],"v),":[29],"\\\\":[30,90],"v_t=\\nabla\\cdot((v+1)^{m_2-1}\\nabla":[31],"v)+\\nabla\\cdot(u\\nabla":[32],"v)-v+u,":[33],"\\end{cases}":[34,95],"\\end{equation}":[35,96],"a":[37],"smooth":[38],"bounded":[39],"domain":[40],"$\u03a9\\subseteq\\mathbb{R}^n$,":[41],"$n\\in\\mathbb{N}$":[42],"nonnegative":[44],"initial":[45],"data":[46],"homogenuous":[48],"Neumann":[49],"boundary":[50],"conditions.":[51],"Moreover,":[52],"we":[53,103],"boundedness":[55,73],"$u$":[57],"$L^\\infty((0,\\infty);\\:L^{m_1}(\u03a9))$":[59],"$v$":[61],"$L^\\infty(\u03a9\\times(0,\\infty))$.":[63],"The":[64],"strategy":[65],"will":[66],"be":[67],"classical":[75],"$(u_\\varepsilon,v_\\varepsilon)$":[77],"approximating":[80],"u_{\\varepsilon":[84],"t}=\\nabla\\cdot((u_\\varepsilon+1)^{m_1-1}\\nabla":[85],"u_\\varepsilon)":[86],"-\\nabla\\cdot(\\frac{u_\\varepsilon\\nabla":[87],"v_\\varepsilon}{1+\\varepsilon\\vert\\nabla":[88,133],"v_\\varepsilon\\vert}),":[89],"v_{\\varepsilon":[91],"t}=\\nabla\\cdot((v_\\varepsilon+1)^{m_2-1}\\nabla":[92],"v_\\varepsilon)+\\nabla\\cdot(u_\\varepsilon\\nabla":[93],"v_\\varepsilon)-v_\\varepsilon+u_\\varepsilon.":[94],"Passing":[97],"limit":[100],"$\\varepsilon\\searrow":[101],"0$,":[102],"obtain":[104],"convergence":[105,130],"second":[111],"first":[117],"system.":[118],"One":[119],"main":[122],"challenges":[123],"lie":[124],"proof":[127],"$\\frac{\\nabla":[132],"v_\\varepsilon\\vert}$":[134],"$\\nabla":[136],"v$":[137],"$L^2(\u03a9\\times(0,T))$":[139],"for":[140],"$T\\in(0,\\infty)$.":[141]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-10T00:00:00"}
