1,024 research outputs found
Eigenelements of a General Aggregation-Fragmentation Model
We consider a linear integro-differential equation which arises to describe
both aggregation-fragmentation processes and cell division. We prove the
existence of a solution (\lb,\U,\phi) to the related eigenproblem. Such
eigenelements are useful to study the long time asymptotic behaviour of
solutions as well as the steady states when the equation is coupled with an
ODE. Our study concerns a non-constant transport term that can vanish at
since it seems to be relevant to describe some biological processes like
proteins aggregation. Non lower-bounded transport terms bring difficulties to
find estimates. All the work of this paper is to solve this problem
using weighted-norms
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
Optimal Regularizing Effect for Scalar Conservation Laws
We investigate the regularity of bounded weak solutions of scalar
conservation laws with uniformly convex flux in space dimension one, satisfying
an entropy condition with entropy production term that is a signed Radon
measure. The proof is based on the kinetic formulation of scalar conservation
laws and on an interaction estimate in physical space.Comment: 24 pages, assumption (11) in Theorem 3.1 modified together with the
example on p. 7, one remark added after the proof of Lemma 4.3, some typos
correcte
Free boundary problems for Tumor Growth: a Viscosity solutions approach
The mathematical modeling of tumor growth leads to singular stiff pressure
law limits for porous medium equations with a source term. Such asymptotic
problems give rise to free boundaries, which, in the absence of active motion,
are generalized Hele-Shaw flows. In this note we use viscosity solutions
methods to study limits for porous medium-type equations with active motion. We
prove the uniform convergence of the density under fairly general assumptions
on the initial data, thus improving existing results. We also obtain some
additional information/regularity about the propagating interfaces, which, in
view of the discontinuities, can nucleate and, thus, change topological type.
The main tool is the construction of local, smooth, radial solutions which
serve as barriers for the existence and uniqueness results as well as to
quantify the speed of propagation of the free boundary propagation
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
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