47,885 research outputs found
The cardiac bidomain model and homogenization
We provide a rather simple proof of a homogenization result for the bidomain
model of cardiac electrophysiology. Departing from a microscopic cellular
model, we apply the theory of two-scale convergence to derive the bidomain
model. To allow for some relevant nonlinear membrane models, we make essential
use of the boundary unfolding operator. There are several complications
preventing the application of standard homogenization results, including the
degenerate temporal structure of the bidomain equations and a nonlinear dynamic
boundary condition on an oscillating surface.Comment: To appear in Networks and Heterogeneous Media, Special Issue on
Mathematical Methods for Systems Biolog
On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis
In this article we deal with a class of strongly coupled parabolic systems
that encompasses two different effects: degenerate diffusion and chemotaxis.
Such classes of equations arise in the mesoscale level modeling of biomass
spreading mechanisms via chemotaxis. We show the existence of an exponential
attractor and, hence, of a finite-dimensional global attractor under certain
'balance conditions' on the order of the degeneracy and the growth of the
chemotactic function
A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains
This paper focuses on rate-independent damage in elastic bodies. Since the
driving energy is nonconvex, solutions may have jumps as a function of time,
and in this situation it is known that the classical concept of energetic
solutions for rate-independent systems may fail to accurately describe the
behavior of the system at jumps. Therefore we resort to the (by now
well-established) vanishing viscosity approach to rate-independent modeling,
and approximate the model by its viscous regularization. In fact, the analysis
of the latter PDE system presents remarkable difficulties, due to its highly
nonlinear character. We tackle it by combining a variational approach to a
class of abstract doubly nonlinear evolution equations, with careful regularity
estimates tailored to this specific system, relying on a q-Laplacian type
gradient regularization of the damage variable. Hence for the viscous problem
we conclude the existence of weak solutions, satisfying a suitable
energy-dissipation inequality that is the starting point for the vanishing
viscosity analysis. The latter leads to the notion of (weak) parameterized
solution to our rate-independent system, which encompasses the influence of
viscosity in the description of the jump regime
On the viscous Cahn-Hilliard equation with singular potential and inertial term
We consider a relaxation of the viscous Cahn-Hilliard equation induced by the
second-order inertial term~. The equation also contains a semilinear
term of "singular" type. Namely, the function is defined only on a
bounded interval of corresponding to the physically admissible
values of the unknown , and diverges as approaches the extrema of that
interval. In view of its interaction with the inertial term , the term
is difficult to be treated mathematically. Based on an approach
originally devised for the strongly damped wave equation, we propose a suitable
concept of weak solution based on duality methods and prove an existence
result.Comment: 11 page
Complete damage in linear elastic materials - Modeling, weak formulation and existence results
In this work, we introduce a degenerating PDE system with a time-depending
domain for complete damage processes under time-varying Dirichlet boundary
conditions. The evolution of the system is described by a doubly nonlinear
differential inclusion for the damage process and a quasi-static balance
equation for the displacement field which are strongly nonlinearly coupled. In
our proposed model, the material may completely disintegrate which is
indispensable for a realistic modeling of damage processes in elastic
materials. Complete damage theories lead to several mathematical problems since
for instance coercivity properties of the free energy are lost and, therefore,
several difficulties arise. For the introduced complete damage model, we
propose a classical formulation and a corresponding suitable weak formulation
in an -framework. The main aim is to prove existence of weak solutions for
the introduced degenerating model. In addition, we show that the classical
differential inclusion can be regained from the notion of weak solutions under
certain regularity assumptions which is a novelty in the theory of complete
damage models of this type
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