Aggregation Equation and Collapse to Singular Measure

Taoufik Hmidi; Dong Li

doi:10.1007/978-3-031-14268-0_4

Aggregation Equation and Collapse to Singular Measure

Taoufik Hmidi, Dong Li

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Abstract

We are concerned with the dynamics of onefold symmetric patches for the two-dimensional aggregation equation associated with the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in subcritical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows to analyze the concentration phenomenon of the aggregation patches near the blowup time. In particular, we prove that the patch collapses to a collection of disjoint segments, and we provide a description of the singular measure through a careful study of the asymptotic behavior of the graph.

Original language	English (US)
Title of host publication	Tutorials, Schools, and Workshops in the Mathematical Sciences
Publisher	Birkhauser
Pages	123-149
Number of pages	27
DOIs	https://doi.org/10.1007/978-3-031-14268-0_4
State	Published - 2022

Publication series

Name	Tutorials, Schools, and Workshops in the Mathematical Sciences
ISSN (Print)	2522-0969
ISSN (Electronic)	2522-0977

Keywords

Aggregation equation
Asymptotic behavior
Concentration phenomenon

ASJC Scopus subject areas

General Mathematics
Physics and Astronomy (miscellaneous)

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10.1007/978-3-031-14268-0_4

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title = "Aggregation Equation and Collapse to Singular Measure",

abstract = "We are concerned with the dynamics of onefold symmetric patches for the two-dimensional aggregation equation associated with the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in subcritical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows to analyze the concentration phenomenon of the aggregation patches near the blowup time. In particular, we prove that the patch collapses to a collection of disjoint segments, and we provide a description of the singular measure through a careful study of the asymptotic behavior of the graph.",

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author = "Taoufik Hmidi and Dong Li",

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T1 - Aggregation Equation and Collapse to Singular Measure

AU - Hmidi, Taoufik

AU - Li, Dong

PY - 2022

Y1 - 2022

N2 - We are concerned with the dynamics of onefold symmetric patches for the two-dimensional aggregation equation associated with the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in subcritical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows to analyze the concentration phenomenon of the aggregation patches near the blowup time. In particular, we prove that the patch collapses to a collection of disjoint segments, and we provide a description of the singular measure through a careful study of the asymptotic behavior of the graph.

AB - We are concerned with the dynamics of onefold symmetric patches for the two-dimensional aggregation equation associated with the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in subcritical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows to analyze the concentration phenomenon of the aggregation patches near the blowup time. In particular, we prove that the patch collapses to a collection of disjoint segments, and we provide a description of the singular measure through a careful study of the asymptotic behavior of the graph.

KW - Aggregation equation

KW - Asymptotic behavior

KW - Concentration phenomenon

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M3 - Chapter

AN - SCOPUS:85142095956

T3 - Tutorials, Schools, and Workshops in the Mathematical Sciences

SP - 123

EP - 149

BT - Tutorials, Schools, and Workshops in the Mathematical Sciences

PB - Birkhauser

ER -

Aggregation Equation and Collapse to Singular Measure

Abstract

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Keywords

ASJC Scopus subject areas

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