Calculus of Variations and Geometric Measure Theory
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G. A. Bonaschi - J. A. Carrillo - M. Di Francesco - M. Peletier

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

created by difrancesco on 15 Oct 2013


Submitted Paper

Inserted: 15 oct 2013
Last Updated: 15 oct 2013

Year: 2013
Links: PDF


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 L2 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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