On the minimizing movement with the 1-Wasserstein distance

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.