32 research outputs found
On cross-diffusion systemsfor two populations subject to a common congestion effect
In this paper, we investigate the existence of solution for systems of
Fokker-Planck equations coupled through a common nonlinear congestion. Two
different kinds of congestion are considered: a porous media congestion or
\textit{soft} congestion and the \textit{hard} congestion given by the
constraint . We show that these systems can be seen
as gradient flows in a Wasserstein product space and then we obtain a
constructive method to prove the existence of solutions. Therefore it is
natural to apply it for numerical purposes and some numerical simulations are
included
On systems of continuity equations with nonlinear diffusion and nonlocal drifts
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
A differential approach to the multi-marginal Schrödinger system
We develop an elementary and self-contained differential approach, in an L ∞ setting, for well-posedness (existence, uniqueness and smooth dependence with respect to the data) for the multi-marginal Schrödinger system which arises in the entropic regularization of optimal transport problems
A splitting method for nonlinear diffusions with nonlocal, nonpotential drifts
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
An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems
International audienceIn this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric framework. We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations. We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical simulations are included