Blow-up phenomena for gradient flows of discrete homogeneous functionals

TL;DR

This paper studies the finite-time blow-up of solutions in gradient flows of homogeneous functionals, revealing that all solutions become singular in finite time, with numerical simulations illustrating complex dynamics.

Contribution

It proves that solutions to certain homogeneous gradient flows inevitably become singular in finite time, extending understanding of blow-up phenomena in particle systems.

Findings

All solutions become singular in finite time.

Numerical simulations show complex nonlinear dynamics.

Solutions with positive energy exhibit striking behaviors.

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.