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.

DOI: 10.1051/cocv/2014032

Keywords: Wasserstein gradient flows, nonlocal interaction equations, entropy solutions, scalar conservation laws, particle approximation

^{1, 2}; Carrillo, José A.

^{3}; Di Francesco, Marco

^{4}; Peletier, Mark A.

^{4}

@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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