Article (Scientific journals)
Blow-Up Phenomena for Gradient Flows of Discrete Homogeneous Functionals
Calvez, V.; Gallouët, Thomas
2019In Applied Mathematics and Optimization, 79 (2), p. 453-481
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Keywords :
Blow up; Degenerate Keller–Segel equations; Dichotomy theorem; Discrete homogeneous functionals; Drift diffusion equations; Mathematical techniques; Blow-up; Diffusing particles; Driftdiffusion equations; Functionals; Non-linear dynamics; Positive energies; Self-interactions; Control engineering
Abstract :
[en] We investigate gradient flows of some homogeneous functionals in R N , arising in the Lagrangian approximation of systems of self-interacting and diffusing particles. We focus on the case of negative homogeneity. In the case of strong self-interaction (super critical case), the functional possesses a cone of negative energy. It is immediate to see that solutions with negative energy at some time become singular in finite time, meaning that a subset of particles concentrate at a single point. Here, we establish that all solutions become singular in finite time, in the super critical case, for the class of functionals under consideration. The paper is completed with numerical simulations illustrating the striking non linear dynamics when initial data have positive energy. © 2017, Springer Science+Business Media, LLC.
Disciplines :
Mathematics
Author, co-author :
Calvez, V.;  Institut Camille Jordan, UMR 5208 CNRS/Universit Claude Bernard Lyon 1 and Project-Team Inria NUMED, Lyon, France
Gallouët, Thomas ;  Université de Liège - ULg
Language :
English
Title :
Blow-Up Phenomena for Gradient Flows of Discrete Homogeneous Functionals
Publication date :
2019
Journal title :
Applied Mathematics and Optimization
ISSN :
0095-4616
eISSN :
1432-0606
Publisher :
Springer, Germany
Volume :
79
Issue :
2
Pages :
453-481
Peer reviewed :
Peer Reviewed verified by ORBi
Available on ORBi :
since 20 December 2021

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