106 research outputs found
The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system
We consider an Allen-Cahn type equation with a bistable nonlinearity
associated to a double-well potential whose well-depths can be slightly
unbalanced, and where the coefficient of the nonlinear reaction term is very
small. Given rather general initial data, we perform a rigorous analysis of
both the generation and the motion of interface. More precisely we show that
the solution develops a steep transition layer within a small time, and we
present an optimal estimate for its width. We then consider a class of
reaction-diffusion systems which includes the FitzHugh-Nagumo system as a
special case. Given rather general initial data, we show that the first
component of the solution vector develops a steep transition layer and that all
the results mentioned above remain true for this component
On a Cahn-Hilliard type phase field system related to tumor growth
The paper deals with a phase field system of Cahn-Hilliard type. For positive
viscosity coefficients, the authors prove an existence and uniqueness result
and study the long time behavior of the solution by assuming the nonlinearities
to be rather general. In a more restricted setting, the limit as the viscosity
coefficients tend to zero is investigated as well.Comment: Key words: phase field model, tumor growth, viscous Cahn-Hilliard
equations, well posedness, long-time behavior, asymptotic analysi
Travelling-wave analysis of a model describing tissue degradation by bacteria
We study travelling-wave solutions for a reaction-diffusion system arising as
a model for host-tissue degradation by bacteria. This system consists of a
parabolic equation coupled with an ordinary differential equation. For large
values of the `degradation-rate parameter' solutions are well approximated by
solutions of a Stefan-like free boundary problem, for which travelling-wave
solutions can be found explicitly. Our aim is to prove the existence of
travelling waves for all sufficiently large wave-speeds for the original
reaction-diffusion system and to determine the minimal speed. We prove that for
all sufficiently large degradation rates the minimal speed is identical to the
minimal speed of the limit problem. In particular, in this parameter range,
nonlinear selection of the minimal speed occurs.Comment: 15 pages, 3 figure
Interface dynamics of the porous medium equation with a bistable reaction term
We consider a degenerate partial differential equation arising in population
dynamics, namely the porous medium equation with a bistable reaction term. We
study its asymptotic behavior as a small parameter, related to the thickness of
a diffuse interface, tends to zero. We prove the rapid formation of transition
layers which then propagate. We prove the convergence to a sharp interface
limit whose normal velocity, at each point, is that of the underlying
degenerate travelling wave
A hyperbolic-elliptic-parabolic PDE model describing chemotactic E. coli colonies
We study a modified version of an initial-boundary value problem describing
the formation of colony patterns of bacteria \textit{Escherichia Coli}. The
original system of three parabolic equations was studied numerically and
analytically and gave insights into the underlying mechanisms of chemotaxis. We
focus here on the parabolic-elliptic-parabolic approximation and the
hyperbolic-elliptic-parabolic limiting system which describes the case of pure
chemotactic movement without random diffusion. We first construct local-in-time
solutions for the parabolic-elliptic-parabolic system. Then we prove uniform
\textit{a priori} estimates and we use them along with a compactness argument
in order to construct local-in-time solutions for the
hyperbolic-elliptic-parabolic limiting system. Finally, we prove that some
initial conditions give rise to solutions which blow-up in finite time in the
norm in all space dimensions. This last result violet is true even
in space dimension 1, which is not the case for the full parabolic or
parabolic-elliptic Keller-Segel systems
Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure
We consider an immiscible incompressible two-phase flow in a porous medium composed of two different rocks so that the capillary pressure field is discontinuous at the interface between the rocks. This leads us to apply a concept of multi-valued phase pressures and a notion of weak solution for the flow which have been introduced in [Cancés \& Pierre, {\em SIAM J. Math. Anal}, 44(2):966--992, 2012]. We discretize the problem by means of a numerical algorithm which reduces to a standard finite volume scheme in each rock and prove the convergence of the approximate solution to a weak solution of the two-phase flow problem. The numerical experiments show in particular that this scheme permits to reproduce the oil trapping phenomenon
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