Geodesic
Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D
Bonaschi, Giovanni A.1, 2 ; Carrillo, José A.3 ; Di Francesco, Marco4 ; 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

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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 :
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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     author = {Bonaschi, Giovanni A. and Carrillo, Jos\'e A. and Di Francesco, Marco and Peletier, Mark A.},
     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},
     number = {2},
     year = {2015},
     doi = {10.1051/cocv/2014032},
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     zbl = {1316.35077},
     language = {en},
     url = {http://geodesic.mathdoc.fr/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

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