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 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.
DOI: 10.1051/cocv/2014032
Keywords: Wasserstein gradient flows, nonlocal interaction equations, entropy solutions, scalar conservation laws, particle approximation
@article{COCV_2015__21_2_414_0, 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}, mrnumber = {3348406}, zbl = {1316.35077}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014032/} }
TY - JOUR AU - Bonaschi, Giovanni A. AU - Carrillo, José A. AU - Di Francesco, Marco AU - Peletier, Mark A. TI - Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 414 EP - 441 VL - 21 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014032/ DO - 10.1051/cocv/2014032 LA - en ID - COCV_2015__21_2_414_0 ER -
%0 Journal Article %A Bonaschi, Giovanni A. %A Carrillo, José A. %A Di Francesco, Marco %A Peletier, Mark A. %T Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 414-441 %V 21 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014032/ %R 10.1051/cocv/2014032 %G en %F COCV_2015__21_2_414_0
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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