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Blow-up dynamics of self-attracting diffusive particles driven by competing convexities

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  • In this paper, we analyze the dynamics of an $N$ particles system evolving according the gradient flow of an energy functional. The particle system is an approximation of the Lagrangian formulation of a one parameter family of non-local drift-diffusion equations in one spatial dimension. We shall prove the global in time existence of the trajectories of the particles (under a sufficient condition on the initial distribution) and give two blow-up criteria. All these results are consequences of the competition between the discrete entropy and the discrete interaction energy. They are also consistent with the continuous setting, that in turn is a one dimension reformulation of the parabolic-elliptic Keller-Segel system in high dimensions.
    Mathematics Subject Classification: 35B44, 35D30, 35Q92, 35K55, 65M99, 92C17, 92B05.


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