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

Bonaschi, Giovanni A.; Carrillo, José A.; Di Francesco, Marco; Peletier, Mark A.

doi:10.1051/cocv/2014032

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.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 414-441.

Résumé

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.

Reçu le : 2013-10-21

MR Zbl

DOI : 10.1051/cocv/2014032

Classification : 35A02, 35F20, 45K05, 35L65, 70F45, 92D25
Mots-clés : Wasserstein gradient flows, nonlocal interaction equations, entropy solutions, scalar conservation laws, particle approximation

Affiliations des auteurs :

Bonaschi, Giovanni A. ^{1, 2} ; Carrillo, José A. ³ ; Di Francesco, Marco ⁴ ; Peletier, Mark A. ⁴

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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},
     volume = {21},
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     mrnumber = {3348406},
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     url = {http://archive.numdam.org/articles/10.1051/cocv/2014032/}
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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, Tome 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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