{"id":"https://openalex.org/W7220738691","doi":"https://doi.org/10.48550/arxiv.2610.06544","title":"Finite-time blowup of the IPM equation from smooth initial data without forcing","display_name":"Finite-time blowup of the IPM equation from smooth initial data without forcing","publication_year":2026,"publication_date":"2026-10-05","ids":{"openalex":"https://openalex.org/W7220738691","doi":"https://doi.org/10.48550/arxiv.2610.06544"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.06544","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06544","pdf_url":"https://arxiv.org/pdf/2610.06544","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/2610.06544","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102882403","display_name":"Fan Zheng","orcid":"https://orcid.org/0000-0002-5718-4010","observed_orcids":["https://orcid.org/0000-0002-5718-4010"]},"institutions":[{"id":"https://openalex.org/I4210157032","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05e9bn444","country_code":"ES","type":"facility","lineage":["https://openalex.org/I4210157032"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Zheng, Fan","raw_affiliation_strings":["Instituto de Ciencias Matem\u00e1ticas"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Instituto de Ciencias Matem\u00e1ticas","institution_ids":["https://openalex.org/I4210157032"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5102882403"],"corresponding_institution_ids":["https://openalex.org/I4210157032"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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/T10940","display_name":"Navier-Stokes equation solutions","score":0.9818999767303467,"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":0.9818999767303467,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.005499999970197678,"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":0.004399999976158142,"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/two-dimensional-torus","display_name":"two-dimensional torus","score":0.906000018119812},{"id":"https://openalex.org/keywords/pressure-hessian","display_name":"pressure Hessian","score":0.5870000123977661},{"id":"https://openalex.org/keywords/smooth-initial-data","display_name":"smooth initial data","score":0.5270000100135803},{"id":"https://openalex.org/keywords/nonlinear-parabolic-equations","display_name":"nonlinear parabolic equations","score":0.44600000977516174}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7404000163078308},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6660000085830688},{"id":"https://openalex.org/C98156149","wikidata":"https://www.wikidata.org/wiki/Q2414143","display_name":"Pressure gradient","level":2,"score":0.5954999923706055},{"id":"https://openalex.org/C197115733","wikidata":"https://www.wikidata.org/wiki/Q1003136","display_name":"Forcing (mathematics)","level":2,"score":0.595300018787384},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5235999822616577},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.4973999857902527},{"id":"https://openalex.org/C84655787","wikidata":"https://www.wikidata.org/wiki/Q8067817","display_name":"Compressibility","level":2,"score":0.4936999976634979},{"id":"https://openalex.org/C17685861","wikidata":"https://www.wikidata.org/wiki/Q6960532","display_name":"Time derivative","level":2,"score":0.4927999973297119},{"id":"https://openalex.org/C107551265","wikidata":"https://www.wikidata.org/wiki/Q1458245","display_name":"Displacement (psychology)","level":2,"score":0.4666999876499176},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4390999972820282},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.40560001134872437},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.38769999146461487},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.385699987411499},{"id":"https://openalex.org/C180188523","wikidata":"https://www.wikidata.org/wiki/Q574576","display_name":"Exponential decay","level":2,"score":0.35370001196861267},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.3508000075817108},{"id":"https://openalex.org/C157709441","wikidata":"https://www.wikidata.org/wiki/Q1411887","display_name":"Uniform convergence","level":3,"score":0.3393000066280365},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.3377000093460083},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.33070001006126404},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3305000066757202},{"id":"https://openalex.org/C2777742833","wikidata":"https://www.wikidata.org/wiki/Q1964083","display_name":"Reciprocal","level":2,"score":0.32280001044273376},{"id":"https://openalex.org/C2780759522","wikidata":"https://www.wikidata.org/wiki/Q7922828","display_name":"Vertical displacement","level":2,"score":0.3206999897956848},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.3192000091075897},{"id":"https://openalex.org/C105569014","wikidata":"https://www.wikidata.org/wiki/Q3271208","display_name":"Porous medium","level":3,"score":0.3151000142097473},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.3125999867916107},{"id":"https://openalex.org/C129747778","wikidata":"https://www.wikidata.org/wiki/Q1014918","display_name":"Burgers' equation","level":3,"score":0.3102000057697296},{"id":"https://openalex.org/C50415386","wikidata":"https://www.wikidata.org/wiki/Q1663717","display_name":"Incompressible flow","level":3,"score":0.3098999857902527},{"id":"https://openalex.org/C27851653","wikidata":"https://www.wikidata.org/wiki/Q829653","display_name":"Bounded variation","level":2,"score":0.30390000343322754},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.30379998683929443},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3034999966621399},{"id":"https://openalex.org/C180205008","wikidata":"https://www.wikidata.org/wiki/Q159190","display_name":"Amplitude","level":2,"score":0.30329999327659607},{"id":"https://openalex.org/C2779664074","wikidata":"https://www.wikidata.org/wiki/Q3518405","display_name":"Terminal (telecommunication)","level":2,"score":0.30070000886917114},{"id":"https://openalex.org/C154954056","wikidata":"https://www.wikidata.org/wiki/Q1604164","display_name":"Transverse plane","level":2,"score":0.2946999967098236},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.29440000653266907},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.28029999136924744},{"id":"https://openalex.org/C111915175","wikidata":"https://www.wikidata.org/wiki/Q1801562","display_name":"Material derivative","level":2,"score":0.2685999870300293},{"id":"https://openalex.org/C11135259","wikidata":"https://www.wikidata.org/wiki/Q2477027","display_name":"Length scale","level":2,"score":0.257999986410141},{"id":"https://openalex.org/C2779227376","wikidata":"https://www.wikidata.org/wiki/Q6505497","display_name":"Layer (electronics)","level":2,"score":0.25679999589920044},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.2563999891281128}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.06544","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06544","pdf_url":"https://arxiv.org/pdf/2610.06544","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.2610.06544","doi":"https://doi.org/10.48550/arxiv.2610.06544","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.06544","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.06544","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06544","pdf_url":"https://arxiv.org/pdf/2610.06544","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":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220738691.pdf","grobid_xml":"https://content.openalex.org/works/W7220738691.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W55957109","https://openalex.org/W1892906362","https://openalex.org/W1958272776","https://openalex.org/W1974569132","https://openalex.org/W1996952201","https://openalex.org/W2006799790","https://openalex.org/W2012201610","https://openalex.org/W2014792425","https://openalex.org/W2038286776","https://openalex.org/W2045087740","https://openalex.org/W2077731142","https://openalex.org/W2129822331","https://openalex.org/W2156818426","https://openalex.org/W2516843855","https://openalex.org/W2798913155","https://openalex.org/W2936074542","https://openalex.org/W2953252600","https://openalex.org/W2963616889","https://openalex.org/W2982427887","https://openalex.org/W4306313011","https://openalex.org/W4313325021","https://openalex.org/W4386839789","https://openalex.org/W4388110353","https://openalex.org/W4389481096","https://openalex.org/W4392769241","https://openalex.org/W4404390467","https://openalex.org/W4406091708","https://openalex.org/W4410803861","https://openalex.org/W4411399925","https://openalex.org/W4412075297","https://openalex.org/W4414880234","https://openalex.org/W4416098666","https://openalex.org/W4416440239","https://openalex.org/W7135015077","https://openalex.org/W7160844870","https://openalex.org/W7161236379","https://openalex.org/W7161773235","https://openalex.org/W7213430106","https://openalex.org/W7220409167"],"related_works":["https://openalex.org/W7220730937","https://openalex.org/W4409956380","https://openalex.org/W7213430106","https://openalex.org/W4404390467","https://openalex.org/W4313325021","https://openalex.org/W2790171425","https://openalex.org/W4414880234","https://openalex.org/W7220409167","https://openalex.org/W4411399925","https://openalex.org/W2599476439"],"abstract_inverted_index":{"We":[0],"prove":[1],"finite-time":[2],"gradient":[3,49,98],"blow-up":[4,23,134],"for":[5,89],"the":[6,13,17,21,43,51,76,90,97,140],"unforced":[7,127],"incompressible":[8],"porous":[9],"media":[10],"equation":[11],"on":[12],"two-dimensional":[14],"torus,":[15],"removing":[16],"forcing":[18],"used":[19],"in":[20,75,148],"IPM":[22],"constructions":[24],"of":[25,46,96,135],"Alp\u00f6ge,":[26],"Buckmaster,":[27],"and":[28,31,33,66,117,142],"Coiculescu":[29],"\\cite{lit:ABC}":[30],"C\u00f3rdoba":[32],"Mart\u00ednez-Zoroa":[34],"\\cite{lit:CMIPM}.":[35],"The":[36,129],"key":[37],"is":[38,102,122],"an":[39,125],"identity":[40],"that":[41],"expresses":[42],"logarithmic":[44],"growth":[45],"a":[47,68,72,85,113,118,145],"layer's":[48],"as":[50],"negative":[52],"pressure":[53,91],"curvature":[54],"transverse":[55],"to":[56,124,144],"its":[57,107],"transported":[58],"phase.":[59],"A":[60],"nearly":[61],"vertical":[62],"parent":[63,78],"layer":[64,101],"rotates":[65],"amplifies":[67],"new":[69],"phase,":[70],"with":[71],"gain":[73],"exponential":[74],"reciprocal":[77],"tilt.":[79],"An":[80],"asymmetric":[81],"periodic":[82],"profile":[83],"gives":[84],"uniform":[86],"upper":[87],"bound":[88],"Hessian,":[92],"preventing":[93],"rapid":[94],"decay":[95],"magnitude.":[99],"Each":[100],"seeded":[103],"at":[104,139],"time":[105],"zero;":[106],"phase":[108],"means":[109],"are":[110],"selected":[111],"by":[112],"terminal":[114,146],"displacement":[115],"problem,":[116],"finite":[119],"order":[120],"expansion":[121],"corrected":[123],"exact":[126],"solution.":[128],"scale":[130],"induction":[131],"then":[132],"shows":[133],"two":[136],"derivative":[137],"components":[138],"origin":[141],"convergence":[143],"density":[147],"every":[149],"$C^\u03b7$,":[150],"$\u03b7<1$.":[151]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-10-07T00:00:00"}