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On systems of continuity equations with nonlinear diffusion and nonlocal drifts

Carlier, Guillaume; Laborde, Maxime (2015-06), On systems of continuity equations with nonlinear diffusion and nonlocal drifts. https://basepub.dauphine.fr/handle/123456789/16911

Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-01147666
Date
2015-06
Series title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Pages
26
Metadata
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Author(s)
Carlier, Guillaume
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Laborde, Maxime
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
This paper is devoted to existence and uniqueness results for classes of nonlinear diffusion equations (or systems) which may be viewed as regular perturbations of Wasserstein gradient flows. First, in the case where the drift is a gradient (in the physical space), we obtain existence by a semi-implicit Jordan-Kinderlehrer-Otto scheme. Then, in the nonpotential case, we derive existence from a regularization procedure and parabolic energy estimates. We also address the uniqueness issue by a displacement convexity argument.
Subjects / Keywords
Wasserstein gradient flows; systems; interacting species; nonlinear diffusion; semi-implicit JKO scheme; nonlinear parabolic equations