Blow-up phenomena for gradient flows of discrete homogeneous functionals

Vincent Calvez (1, 2, 3) , Thomas O. Gallouët (4)

1 NUMED - Numerical Medicine
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
3 MMCS - Modélisation mathématique, calcul scientifique
4 CMLS - Centre de Mathématiques Laurent Schwartz

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.

Mots clés: discrete homogeneous functionals; dichotomy theorem; degenerate Keller-Segel equations; drift diffusion equations; Blow up

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Analyse numérique [math.NA]; Systèmes dynamiques [math.DS]

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⟩