Abstract
We investigate gradient flows of some homogeneous functionals in \(\mathbb R^N\), 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.
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Funding
This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (Grant Agreement No. 639638). T. O. Gallouët was supported by the ANR contract ISOTACE (ANR-12-MONU-013) and by the Fonds de la Recherche Scientifique - FNRS under Grant MIS F.4539.16.
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Calvez, V., Gallouët, T.O. Blow-Up Phenomena for Gradient Flows of Discrete Homogeneous Functionals. Appl Math Optim 79, 453–481 (2019). https://doi.org/10.1007/s00245-017-9443-z
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DOI: https://doi.org/10.1007/s00245-017-9443-z
Keywords
- Blow up
- Dichotomy theorem
- Degenerate Keller–Segel equations
- Discrete homogeneous functionals
- Drift diffusion equations