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arXiv · 1310.4110

Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

Giovanni A. Bonaschi·José A. Carrillo·Marco Di Francesco·Mark A. Peletier
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Abstract

We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the $L^2$ gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the sub-differential of the functional involved in the Wasserstein gradient flow.

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Giovanni A. Bonaschi, José A. Carrillo, Marco Di Francesco, Mark A. Peletier. 2013-10-15. Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D. https://arxiv.org/abs/1310.4110

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