40 research outputs found
Incompressible immiscible multiphase flows in porous media: a variational approach
We describe the competitive motion of (N + 1) incompressible immiscible
phases within a porous medium as the gradient flow of a singular energy in the
space of non-negative measures with prescribed mass endowed with some tensorial
Wasserstein distance. We show the convergence of the approximation obtained by
a minimization schem\`e a la [R. Jordan, D. Kinder-lehrer \& F. Otto, SIAM J.
Math. Anal, 29(1):1--17, 1998]. This allow to obtain a new existence result for
a physically well-established system of PDEs consisting in the Darcy-Muskat law
for each phase, N capillary pressure relations, and a constraint on the volume
occupied by the fluid. Our study does not require the introduction of any
global or complementary pressure
On the convexity of injectivity domains on nonfocal manifolds
Given a smooth nonfocal compact Riemannian manifold, we show that the
so-called Ma--Trudinger--Wang condition implies the convexity of injectivity
domains. This improves a previous result by Loeper and Villani
The gradient flow structure for incompressible immiscible two-phase flows in porous media
We show that the widely used model governing the motion of two incompressible
immiscible fluids in a possibly heterogeneous porous medium has a formal
gradient flow structure. More precisely, the fluid composition is governed by
the gradient flow of some non-smooth energy. Starting from this energy together
with a dissipation potential, we recover the celebrated Darcy-Muskat law and
the capillary pressure law governing the flow thanks to the principle of least
action. Our interpretation does not require the introduction of any algebraic
transformation like, e.g., the global pressure or the Kirchhoff transform, and
can be transposed to the case of more phases
Blow-up phenomena for gradient flows of discrete homogeneous functionals
International audienceWe investigate gradient flows of some homogeneous functionals in R^N , arising in the Lagrangian approximation of systems of self-interacting and diffusing particles. We focus on the case of negative homogeneity. In the case of strong self-interaction, the functional possesses a cone of negative energy. It is immediate to see that solutions with negative energy at some time become singular in finite time, meaning that a subset of particles concentrate at a single point. Here, we establish that all solutions become singular in finite time for the class of functionals under consideration. The paper is completed with numerical simulations illustrating the striking non linear dynamics when initial data have positive energy
Second order models for optimal transport and cubic splines on the Wasserstein space
On the space of probability densities, we extend the Wasserstein geodesics to
the case of higher-order interpolation such as cubic spline interpolation.
After presenting the natural extension of cubic splines to the Wasserstein
space, we propose a simpler approach based on the relaxation of the variational
problem on the path space. We explore two different numerical approaches, one
based on multi-marginal optimal transport and entropic regularization and the
other based on semi-discrete optimal transport
A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows
In this article we set up a splitting variant of the JKO scheme in order to
handle gradient flows with respect to the Kantorovich-Fisher-Rao metric,
recently introduced and defined on the space of positive Radon measure with
varying masses. We perform successively a time step for the quadratic
Wasserstein/Monge-Kantorovich distance, and then for the Hellinger/Fisher-Rao
distance. Exploiting some inf-convolution structure of the metric we show
convergence of the whole process for the standard class of energy functionals
under suitable compactness assumptions, and investigate in details the case of
internal energies. The interest is double: On the one hand we prove existence
of weak solutions for a certain class of reaction-advection-diffusion
equations, and on the other hand this process is constructive and well adapted
to available numerical solvers.Comment: Final version, to appear in SIAM SIM
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
A variational finite volume scheme for Wasserstein gradient flows
International audienceWe propose a variational finite volume scheme to approximate the solutions to Wasserstein gradient flows. The time discretization is based on an implicit linearization of the Wasserstein distance expressed thanks to Benamou-Brenier formula, whereas space discretization relies on upstream mobility two-point flux approximation finite volumes. Our scheme is based on a first discretize then optimize approach in order to preserve the variational structure of the continuous model at the discrete level. Our scheme can be applied to a wide range of energies, guarantees non-negativity of the discrete solutions as well as decay of the energy. We show that our scheme admits a unique solution whatever the convex energy involved in the continuous problem , and we prove its convergence in the case of the linear Fokker-Planck equation with positive initial density. Numerical illustrations show that it is first order accurate in both time and space, and robust with respect to both the energy and the initial profile