Analysis and mean-field derivation of a porous-medium equation with fractional diffusion

A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interprete...

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Published inCommunications in partial differential equations Vol. 47; no. 11; pp. 2217 - 2269
Main Authors Chen, Li, Holzinger, Alexandra, Jüngel, Ansgar, Zamponi, Nicola
Format Journal Article
LanguageEnglish
Published Philadelphia Taylor & Francis 02.11.2022
Taylor & Francis Ltd
Subjects
Diffusion
Estimates
Existence analysis
fractional diffusion
Inequalities
interacting particle systems
Mathematical analysis
mean-field limit
nonlocal porous-medium equation
Porous media
propagation of chaos
Regularization
Transport equations
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ISSN0360-5302
1532-4133
DOI10.1080/03605302.2022.2118608

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Abstract A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger's approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo-Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.
AbstractList A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger's approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo-Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.
Author Holzinger, Alexandra
Jüngel, Ansgar
Zamponi, Nicola
Chen, Li
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  givenname: Nicola
  surname: Zamponi
  fullname: Zamponi, Nicola
  organization: School of Business Informatics and Mathematics, University of Mannheim
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Snippet A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal...
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SubjectTerms Diffusion
Estimates
Existence analysis
fractional diffusion
Inequalities
interacting particle systems
Mathematical analysis
mean-field limit
nonlocal porous-medium equation
Porous media
propagation of chaos
Regularization
Transport equations
Title Analysis and mean-field derivation of a porous-medium equation with fractional diffusion
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