On the minimizing movement with the 1-Wasserstein distance

¹ Department of Mathematics and Statistics, University of Victoria, P.O. Box. 3060 STN CSC, Victoria B.C., V8W 3R4, Canada
² This work was completed after Martial passed away. We wish to dedicate this article to his memory
³ Ceremade, UMR CNRS 7534, Université Paris Dauphine, Pl. de Lattre de Tassigny, 75775, Paris Cedex 16, France, and MOKAPLAN, INRIA-Paris
⁴ Institut de recherche XLIM-DMI, UMR-CNRS 7252, Faculté des Sciences et Techniques, Université de Limoges 123, Avenue Albert Thomas 87060 Limoges, France

^* Corresponding author: carlier@ceremade.dauphine.fr

Received: 18 March 2017
Accepted: 22 August 2017

Abstract

We consider a class of doubly nonlinear constrained evolution equations which may be viewed as a nonlinear extension of the growing sandpile model of [L. Prigozhin, Eur. J. Appl. Math.  7 (1996) 225–235.]. We prove existence of weak solutions for quite irregular sources by a semi-implicit scheme in the spirit of the seminal works of [R. Jordan et al., SIAM J. Math. Anal.  29 (1998) 1–17, D. Kinderlehrer and N.J. Walkington, Math. Model. Numer. Anal.  33 (1999) 837–852.] but with the 1-Wasserstein distance instead of the quadratic one. We also prove an L¹-contraction result when the source is L¹ and deduce uniqueness and stability in this case.