160 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
Linear relaxation to planar Travelling Waves in Inertial Confinement Fusion
We study linear stability of planar travelling waves for a scalar
reaction-diffusion equation with non-linear anisotropic diffusion. The
mathematical model is derived from the full thermo-hydrodynamical model
describing the process of Inertial Confinement Fusion. We show that solutions
of the Cauchy problem with physically relevant initial data become planar
exponentially fast with rate s(\eps',k)>0, where
\eps'=\frac{T_{min}}{T_{max}}\ll 1 is a small temperature ratio and
the transversal wrinkling wavenumber of perturbations. We rigorously recover in
some particular limit (\eps',k)\rightarrow (0,+\infty) a dispersion relation
s(\eps',k)\sim \gamma_0 k^{\alpha} previously computed heuristically and
numerically in some physical models of Inertial Confinement Fusion
A KPP road-field system with spatially periodic exchange terms
We take interest in a reaction-diffusion system which has been recently
proposed [11] as a model for the effect of a road on propagation phenomena
arising in epidemiology and ecology. This system consists in coupling a
classical Fisher-KPP equation in a half-plane with a line with fast diffusion
accounting for a straight road. The effect of the line on spreading properties
of solutions (with compactly supported initial data) was investigated in a
series of works starting from [11]. We recover these earlier results in a more
general spatially periodic framework by exhibiting a threshold for road
diffusion above which the propagation is driven by the road and the global
speed is accelerated. We also discuss further applications of our approach,
which will rely on the construction of a suitable generalized principal
eigenvalue, and investigate in particular the spreading of solutions with
exponentially decaying initial data.Comment: Updated version, minor typos and details fixe
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