Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D
ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 2, pp. 414-441.

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 subdifferential of the functional involved in the Wasserstein gradient flow.

Received:
DOI: 10.1051/cocv/2014032
Classification: 35A02, 35F20, 45K05, 35L65, 70F45, 92D25
Keywords: Wasserstein gradient flows, nonlocal interaction equations, entropy solutions, scalar conservation laws, particle approximation
Bonaschi, Giovanni A. 1, 2; Carrillo, José A. 3; Di Francesco, Marco 4; Peletier, Mark A. 4

1 Institute for Complex Molecular Systems and Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands.
2 Dipartimento di Matematica, Università di Pavia, 27100 Pavia, Italy.
3 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom.
4 Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom.
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     title = {Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in {1D}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {414--441},
     publisher = {EDP-Sciences},
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Bonaschi, Giovanni A.; Carrillo, José A.; Di Francesco, Marco; Peletier, Mark A. Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 2, pp. 414-441. doi : 10.1051/cocv/2014032. http://archive.numdam.org/articles/10.1051/cocv/2014032/

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