Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D
1 Institute for Complex Molecular
Systems and Department of Mathematics and Computer Science, Technische Universiteit
Eindhoven, P.O. Box
513, 5600 MB,
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.
Revised: 24 April 2014
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 L2 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.
Mathematics Subject Classification: 35A02 / 35F20 / 45K05 / 35L65 / 70F45 / 92D25
Key words: Wasserstein gradient flows / nonlocal interaction equations / entropy solutions / scalar conservation laws / particle approximation
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