Blow-up phenomena for gradient flows of discrete homogeneous functionals - Archive ouverte HAL Access content directly
Journal Articles Applied Mathematics and Optimization Year : 2017

Blow-up phenomena for gradient flows of discrete homogeneous functionals

Abstract

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, 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 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.
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Dates and versions

hal-01286518 , version 1 (17-03-2016)
hal-01286518 , version 2 (17-03-2016)

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Vincent Calvez, Thomas O. Gallouët. Blow-up phenomena for gradient flows of discrete homogeneous functionals. Applied Mathematics and Optimization, 2017, ⟨10.1007/s00245-017-9443-z⟩. ⟨hal-01286518v2⟩
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