On the minimizing movement with the 1-Wasserstein distance.

Summary We consider a class of doubly nonlinear constrained evolution equations which may be viewed as a nonlinear extension of the growing sandpile model of [15]. We prove existence of weak solutions for quite irregular sources by a semi-implicit scheme in the spirit of the seminal works of [13] and [14] but with the 1-Wasserstein distance instead of the quadratic one. We also prove an L 1-contraction result when the source is L 1 and deduce uniqueness and stability in this case.