84 research outputs found
Incompressible limit of mechanical model of tumor growth with viscosity
Various models of tumor growth are available in the litterature. A first
class describes the evolution of the cell number density when considered as a
continuous visco-elastic material with growth. A second class, describes the
tumor as a set and rules for the free boundary are given related to the
classical Hele-Shaw model of fluid dynamics. Following the lines of previous
papers where the material is described by a purely elastic material, or when
active cell motion is included, we make the link between the two levels of
description considering the 'stiff pressure law' limit. Even though viscosity
is a regularizing effect, new mathematical difficulties arise in the
visco-elastic case because estimates on the pressure field are weaker and do
not imply immediately compactness. For instance, traveling wave solutions and
numerical simulations show that the pressure may be discontinous in space which
is not the case for the elastic case.Comment: 17 page
On the Inverse Problem for a Size-Structured Population Model
We consider a size-structured model for cell division and address the
question of determining the division (birth) rate from the measured stable size
distribution of the population. We formulate such question as an inverse
problem for an integro-differential equation posed on the half line. We develop
firstly a regular dependency theory for the solution in terms of the
coefficients and, secondly, a novel regularization technique for tackling this
inverse problem which takes into account the specific nature of the equation.
Our results rely also on generalized relative entropy estimates and related
Poincar\'e inequalities
A simple derivation of BV bounds for inhomogeneous relaxation systems
We consider relaxation systems of transport equations with heterogeneous
source terms and with boundary conditions, which limits are scalar conservation
laws. Classical bounds fail in this context and in particular BV estimates.
They are the most standard and simplest way to prove compactness and
convergence. We provide a novel and simple method to obtain partial BV
regularity and strong compactness in this framework. The standard notion of
entropy is not convenient either and we also indicate another, but closely
related, notion. We give two examples motivated by renal flows which consist of
2 by 2 and 3 by 3 relaxation systems with 2-velocities but the method is more
general
Mathematical Analysis of a System for Biological Network Formation
Motivated by recent physics papers describing rules for natural network
formation, we study an elliptic-parabolic system of partial differential
equations proposed by Hu and Cai. The model describes the pressure field thanks
to Darcy's type equation and the dynamics of the conductance network under
pressure force effects with a diffusion rate representing randomness in the
material structure. We prove the existence of global weak solutions and of
local mild solutions and study their long term behaviour. It turns out that, by
energy dissipation, steady states play a central role to understand the pattern
capacity of the system. We show that for a large diffusion coefficient , the
zero steady state is stable. Patterns occur for small values of because the
zero steady state is Turing unstable in this range; for we can exhibit a
large class of dynamically stable (in the linearized sense) steady states
Competition and boundary formation in heterogeneous media: Application to neuronal differentiation
We analyze an inhomogeneous system of coupled reaction-diffusion equations
representing the dynamics of gene expression during differentiation of nerve
cells. The outcome of this developmental phase is the formation of distinct
functional areas separated by sharp and smooth boundaries. It proceeds through
the competition between the expression of two genes whose expression is driven
by monotonic gradients of chemicals, and the products of gene expression
undergo local diffusion and drive gene expression in neighboring cells. The
problem therefore falls in a more general setting of species in competition
within a non-homogeneous medium. We show that in the limit of arbitrarily small
diffusion, there exists a unique monotonic stationary solution, which splits
the neural tissue into two winner-take-all parts at a precise boundary point:
on both sides of the boundary, different neuronal types are present. In order
to further characterize the location of this boundary, we use a blow-up of the
system and define a traveling wave problem parametrized by the position within
the monotonic gradient: the precise boundary location is given by the unique
point in space at which the speed of the wave vanishes
An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations
For monotone linear differential systems with periodic coefficients, the
(first) Floquet eigenvalue measures the growth rate of the system. We define an
appropriate arithmetico-geometric time average of the coefficients for which we
can prove that the Perron eigenvalue is smaller than the Floquet eigenvalue. We
apply this method to Partial Differential Equations, and we use it for an
age-structured systems of equations for the cell cycle. This opposition between
Floquet and Perron eigenvalues models the loss of circadian rhythms by cancer
cells.Comment: 7 pages, in English, with an abridged French versio
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