Non existence and strong ill-posedness in H2 for the stable IPM equation

We prove the non-existence and strong ill-posedness of the Incompressible Porous Media (IPM) equation for initial data that are small H2(R2) perturbations of the linearly stable profile −x2. A remarkable novelty of the proof is the construction of an H2 perturbation, which solves the IPM equation an...

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Autores: Bianchini, R., Córdoba, D., Martínez-Zoroa, L.
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2025
País:España
Institución:Consejo Superior de Investigaciones Científicas (CSIC)
Repositorio:DIGITAL.CSIC. Repositorio Institucional del CSIC
OAI Identifier:oai:digital.csic.es:10261/423208
Acceso en línea:http://hdl.handle.net/10261/423208
https://www.scopus.com/inward/record.uri?eid=2-s2.0-105009014777&doi=10.1016%2Fj.jfa.2025.111097&partnerID=40&md5=2b99d5288a451e93287be95e8f6b1267
Access Level:acceso abierto
Palabra clave:Non-existence and strong ill-posedness
Partial and anisotropic dissipation
Stable IPM equations
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spelling Non existence and strong ill-posedness in H2 for the stable IPM equationBianchini, R.Córdoba, D.Martínez-Zoroa, L.Non-existence and strong ill-posednessPartial and anisotropic dissipationStable IPM equationsWe prove the non-existence and strong ill-posedness of the Incompressible Porous Media (IPM) equation for initial data that are small H2(R2) perturbations of the linearly stable profile −x2. A remarkable novelty of the proof is the construction of an H2 perturbation, which solves the IPM equation and neutralizes the stabilizing effect of the background profile near the origin, where a strong deformation leading to non-existence in H2 is created. This strong deformation is achieved through an iterative procedure inspired by the work of Córdoba and Martínez-Zoroa (2022) [7]. However, several differences - beyond purely technical aspects - arise due to the anisotropic and, more importantly, to the partially dissipative nature of the equation, adding further challenges to the analysis. © 2025 The Author(s)RB is partially supported by the Italian Ministry of University and Research, PRIN 2020 entitled “PDEs, fluid dynamics and transport equation” and PRIN 2022HSSYPN “Turbulent Effects vs Stability in Equations from Oceanography”, PNRR Italia Domani, funded by the European Union under NextGenerationEU, CUP B53D23009300001. RB warmly thanks Instituto de Ciencias Matemáticas, where part of this work was elaborated. This work is supported in part by the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S & CEX2023-001347-S)” and 152878NB-I00. We were also partially supported by the ERC Advanced Grant 788250, and by the SNF grant FLUTURA: Fluids, Turbulence, Advection No. 212573.Peer reviewedElsevierMinisterio de Ciencia e Innovación (España)Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]202620262025info:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501Publisher's versioninfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/10261/423208https://www.scopus.com/inward/record.uri?eid=2-s2.0-105009014777&doi=10.1016%2Fj.jfa.2025.111097&partnerID=40&md5=2b99d5288a451e93287be95e8f6b1267reponame:DIGITAL.CSIC. Repositorio Institucional del CSICinstname:Consejo Superior de Investigaciones Científicas (CSIC)Ingléshttps://doi.org/10.1016/j.jfa.2025.111097Síinfo:eu-repo/semantics/openAccessoai:digital.csic.es:10261/4232082026-05-22T06:33:51Z
dc.title.none.fl_str_mv Non existence and strong ill-posedness in H2 for the stable IPM equation
title Non existence and strong ill-posedness in H2 for the stable IPM equation
spellingShingle Non existence and strong ill-posedness in H2 for the stable IPM equation
Bianchini, R.
Non-existence and strong ill-posedness
Partial and anisotropic dissipation
Stable IPM equations
title_short Non existence and strong ill-posedness in H2 for the stable IPM equation
title_full Non existence and strong ill-posedness in H2 for the stable IPM equation
title_fullStr Non existence and strong ill-posedness in H2 for the stable IPM equation
title_full_unstemmed Non existence and strong ill-posedness in H2 for the stable IPM equation
title_sort Non existence and strong ill-posedness in H2 for the stable IPM equation
dc.creator.none.fl_str_mv Bianchini, R.
Córdoba, D.
Martínez-Zoroa, L.
author Bianchini, R.
author_facet Bianchini, R.
Córdoba, D.
Martínez-Zoroa, L.
author_role author
author2 Córdoba, D.
Martínez-Zoroa, L.
author2_role author
author
dc.contributor.none.fl_str_mv Ministerio de Ciencia e Innovación (España)
Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]
dc.subject.none.fl_str_mv Non-existence and strong ill-posedness
Partial and anisotropic dissipation
Stable IPM equations
topic Non-existence and strong ill-posedness
Partial and anisotropic dissipation
Stable IPM equations
description We prove the non-existence and strong ill-posedness of the Incompressible Porous Media (IPM) equation for initial data that are small H2(R2) perturbations of the linearly stable profile −x2. A remarkable novelty of the proof is the construction of an H2 perturbation, which solves the IPM equation and neutralizes the stabilizing effect of the background profile near the origin, where a strong deformation leading to non-existence in H2 is created. This strong deformation is achieved through an iterative procedure inspired by the work of Córdoba and Martínez-Zoroa (2022) [7]. However, several differences - beyond purely technical aspects - arise due to the anisotropic and, more importantly, to the partially dissipative nature of the equation, adding further challenges to the analysis. © 2025 The Author(s)
publishDate 2025
dc.date.none.fl_str_mv 2025
2026
2026
dc.type.none.fl_str_mv info:eu-repo/semantics/article
http://purl.org/coar/resource_type/c_6501
Publisher's version
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10261/423208
https://www.scopus.com/inward/record.uri?eid=2-s2.0-105009014777&doi=10.1016%2Fj.jfa.2025.111097&partnerID=40&md5=2b99d5288a451e93287be95e8f6b1267
url http://hdl.handle.net/10261/423208
https://www.scopus.com/inward/record.uri?eid=2-s2.0-105009014777&doi=10.1016%2Fj.jfa.2025.111097&partnerID=40&md5=2b99d5288a451e93287be95e8f6b1267
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://doi.org/10.1016/j.jfa.2025.111097
Sí
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:DIGITAL.CSIC. Repositorio Institucional del CSIC
instname:Consejo Superior de Investigaciones Científicas (CSIC)
instname_str Consejo Superior de Investigaciones Científicas (CSIC)
reponame_str DIGITAL.CSIC. Repositorio Institucional del CSIC
collection DIGITAL.CSIC. Repositorio Institucional del CSIC
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