A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators

Sylvia Serfaty; Juan Luis Vázquez

doi:10.1007/s00526-013-0613-9

A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators

Sylvia Serfaty, Juan Luis Vázquez

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a "mean field" equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the "elementary vortex patch".

Original language	English (US)
Pages (from-to)	1091-1120
Number of pages	30
Journal	Calculus of Variations and Partial Differential Equations
Volume	49
Issue number	3-4
DOIs	https://doi.org/10.1007/s00526-013-0613-9
State	Published - Mar 2014

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1007/s00526-013-0613-9

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title = "A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators",

abstract = "In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a {"}mean field{"} equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the {"}elementary vortex patch{"}.",

author = "Sylvia Serfaty and V{\'a}zquez, {Juan Luis}",

note = "Funding Information: We would like to thank J. A. Carrillo for pointing out to us the reference [] after a first draft of this paper was written. S. S. thanks the Universidad Aut{\'o}noma de Madrid for its hospitality that allowed this work to be completed. S. S. was supported by a EURYI award, and JLV by Spanish Project MTM2008-06326-C02-01.",

year = "2014",

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doi = "10.1007/s00526-013-0613-9",

language = "English (US)",

volume = "49",

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journal = "Calculus of Variations and Partial Differential Equations",

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T1 - A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators

AU - Serfaty, Sylvia

AU - Vázquez, Juan Luis

N1 - Funding Information: We would like to thank J. A. Carrillo for pointing out to us the reference [] after a first draft of this paper was written. S. S. thanks the Universidad Autónoma de Madrid for its hospitality that allowed this work to be completed. S. S. was supported by a EURYI award, and JLV by Spanish Project MTM2008-06326-C02-01.

PY - 2014/3

Y1 - 2014/3

N2 - In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a "mean field" equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the "elementary vortex patch".

AB - In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a "mean field" equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the "elementary vortex patch".

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JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 3-4

ER -

A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators

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