7 research outputs found
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
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
Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up
We investigate a particle system which is a discrete and deterministic
approximation of the one-dimensional Keller-Segel equation with a logarithmic
potential. The particle system is derived from the gradient flow of the
homogeneous free energy written in Lagrangian coordinates. We focus on the
description of the blow-up of the particle system, namely: the number of
particles involved in the first aggregate, and the limiting profile of the
rescaled system. We exhibit basins of stability for which the number of
particles is critical, and we prove a weak rigidity result concerning the
rescaled dynamics. This work is complemented with a detailed analysis of the
case where only three particles interact