A splitting method for nonlinear diffusions with nonlocal, nonpotential drifts

Guillaume Carlier (1, 2) , Maxime Laborde (1, 2)

1 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We prove an existence result for nonlinear diffusion equations in the presence of a nonlocal density-dependent drift which is not necessarily potential. The proof is constructive and based on the Helmholtz decomposition of the drift and a splitting scheme. The splitting scheme combines transport steps by the divergence-free part of the drift and semi-implicit minimization steps à la Jordan-Kinderlherer Otto to deal with the potential part.

Mots clés: Helmholtz decomposition; nonlin-ear diffusions; splitting; nonlocal drift; Wasserstein gradient flows; Jordan-Kinderlehrer-Otto scheme

Domaines: Mathématiques [math]

Guillaume Carlier, Maxime Laborde. A splitting method for nonlinear diffusions with nonlocal, nonpotential drifts. 2016. ⟨hal-01332356⟩