Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities
We establish boundedness estimates for solutions of generalized porous medium equations of the form ∂tu+(−L)[um]=0in RN×(0,T), where m≥1 and −L is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/705890 |
| Acceso en línea: | http://hdl.handle.net/10486/705890 https://dx.doi.org/10.1016/j.jfa.2022.109831 |
| Access Level: | acceso abierto |
| Palabra clave: | Boundedness estimates Gagliardo-Nirenberg-Sobolev inequalities Green functions Nonlinear degenerate parabolic equations Matemáticas |
| id |
ES_f99bcf223cdf702c0eee07726f2d0bdf |
|---|---|
| oai_identifier_str |
oai:repositorio.uam.es:10486/705890 |
| network_acronym_str |
ES |
| network_name_str |
España |
| repository_id_str |
|
| spelling |
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalitiesBonforte, MatteoEndal, JørgenBoundedness estimatesGagliardo-Nirenberg-Sobolev inequalitiesGreen functionsNonlinear degenerate parabolic equationsMatemáticasWe establish boundedness estimates for solutions of generalized porous medium equations of the form ∂tu+(−L)[um]=0in RN×(0,T), where m≥1 and −L is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative estimates take the form of precise L1–L∞-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of −L and I−L. In the linear case m=1, it is well-known that the L1–L∞-smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques. We establish a similar scenario in the nonlinear setting m>1. First, we can show that operators for which ultracontractivity holds, also provide L1–L∞-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by 0-order Lévy operators like −L=I−J⁎. They do not regularize when m=1, but we show that surprisingly enough they do so when m>1, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator. Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iterationJ. Endal has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 839749 “Novel techniques for quantitative behaviour of convection-diffusion equations (techFRONT)”, and from the Research Council of Norway under the MSCA-TOPP-UT grant agreement no. 312021. M. Bonforte was partially supported by the Projects MTM2017-85757-P and PID2020- 113596GB-I00 (Spanish Ministry of Science and Innovation). M. Bonforte moreover acknowledges financial support from the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019- 000904-S) and by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 777822ElsevierDepartamento de MatemáticasFacultad de Ciencias20222022-12-27research articlehttp://purl.org/coar/resource_type/c_2df8fbb1VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10486/705890https://dx.doi.org/10.1016/j.jfa.2022.109831reponame:Biblos-e Archivo. Repositorio Institucional de la UAMinstname:Universidad Autónoma de MadridInglésengEuropean Commission http://dx.doi.org/10.13039/501100000780 Horizon 2020 Framework Programme 839749European Commission http://dx.doi.org/10.13039/501100000780 Horizon 2020 Framework Programme 777822open accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:repositorio.uam.es:10486/7058902026-06-23T12:46:27Z |
| dc.title.none.fl_str_mv |
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities |
| title |
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities |
| spellingShingle |
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities Bonforte, Matteo Boundedness estimates Gagliardo-Nirenberg-Sobolev inequalities Green functions Nonlinear degenerate parabolic equations Matemáticas |
| title_short |
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities |
| title_full |
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities |
| title_fullStr |
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities |
| title_full_unstemmed |
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities |
| title_sort |
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities |
| dc.creator.none.fl_str_mv |
Bonforte, Matteo Endal, Jørgen |
| author |
Bonforte, Matteo |
| author_facet |
Bonforte, Matteo Endal, Jørgen |
| author_role |
author |
| author2 |
Endal, Jørgen |
| author2_role |
author |
| dc.contributor.none.fl_str_mv |
Departamento de Matemáticas Facultad de Ciencias |
| dc.subject.none.fl_str_mv |
Boundedness estimates Gagliardo-Nirenberg-Sobolev inequalities Green functions Nonlinear degenerate parabolic equations Matemáticas |
| topic |
Boundedness estimates Gagliardo-Nirenberg-Sobolev inequalities Green functions Nonlinear degenerate parabolic equations Matemáticas |
| description |
We establish boundedness estimates for solutions of generalized porous medium equations of the form ∂tu+(−L)[um]=0in RN×(0,T), where m≥1 and −L is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative estimates take the form of precise L1–L∞-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of −L and I−L. In the linear case m=1, it is well-known that the L1–L∞-smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques. We establish a similar scenario in the nonlinear setting m>1. First, we can show that operators for which ultracontractivity holds, also provide L1–L∞-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by 0-order Lévy operators like −L=I−J⁎. They do not regularize when m=1, but we show that surprisingly enough they do so when m>1, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator. Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iteration |
| publishDate |
2022 |
| dc.date.none.fl_str_mv |
2022 2022-12-27 |
| dc.type.none.fl_str_mv |
research article http://purl.org/coar/resource_type/c_2df8fbb1 VoR http://purl.org/coar/version/c_970fb48d4fbd8a85 |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/article |
| format |
article |
| dc.identifier.none.fl_str_mv |
http://hdl.handle.net/10486/705890 https://dx.doi.org/10.1016/j.jfa.2022.109831 |
| url |
http://hdl.handle.net/10486/705890 https://dx.doi.org/10.1016/j.jfa.2022.109831 |
| dc.language.none.fl_str_mv |
Inglés eng |
| language_invalid_str_mv |
Inglés |
| language |
eng |
| dc.relation.none.fl_str_mv |
European Commission http://dx.doi.org/10.13039/501100000780 Horizon 2020 Framework Programme 839749 European Commission http://dx.doi.org/10.13039/501100000780 Horizon 2020 Framework Programme 777822 |
| dc.rights.none.fl_str_mv |
open access http://purl.org/coar/access_right/c_abf2 |
| dc.rights.openaire.fl_str_mv |
info:eu-repo/semantics/openAccess |
| rights_invalid_str_mv |
open access http://purl.org/coar/access_right/c_abf2 |
| 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:Biblos-e Archivo. Repositorio Institucional de la UAM instname:Universidad Autónoma de Madrid |
| instname_str |
Universidad Autónoma de Madrid |
| reponame_str |
Biblos-e Archivo. Repositorio Institucional de la UAM |
| collection |
Biblos-e Archivo. Repositorio Institucional de la UAM |
| repository.name.fl_str_mv |
|
| repository.mail.fl_str_mv |
|
| _version_ |
1869425110036774912 |
| score |
15.300724 |