6th AMARENA DaYs

Amiens-Milano-poitiers REunion on Numerical and mathematical Analysis

May 3 - 4 2018

6th AMARENA DaYs

Amiens-Milano-poitiers REunion on Numerical and mathematical Analysis

May 3 - 4 2018

Laboratoire Amiénois de Mathématique

Fondamentale et Appliquée

CNRS UMR 7352

PICS CNRS AMaMMO

Program

Salle de séminaires

LAMFA, UPJV

First day: Thursday 3

9h Welcome in LAMFA

9h30-10h15 Maurizio Grasselli (Politecnico Milano)

The Cahn-Hilliard equation and the separation property

10h15-11h Monica Conti (Politecnico Milano)

A viscoelastic equation with nonlinear density and memory effects

11h-11h15 : break

11h15 -12h Alain Miranville (Poitiers)

Mathematical models for brain lactate kinetics

12h-14h : lunch

14h-14h45 Cecilia Cavaterra (Università statale di Milano)

An inverse problem arising from cardiac electrophysiology

15h-15h45 Jean-Paul Chehab (LAMFA, UPJV)

Stabilization schemes for Phase field equations

Second day: Friday 4

9h Welcome in LAMFA

9h30-10h15 Alejandro A. Franco (LRCS, UPJV)

White and Black Box computational approaches for the prediction of better batteries

10h15-11h Filippo Dell'Oro (Politecnico Milano)

Semidiscrete schemes for evolution equations with memory

11h-11h15 break

11h15 -12h Benedetta Noris (LAMFA, UPJV)

Orbital stability of the ground states for the nonlinear Schrödinger equation with harmonic potential.

12h-14h : lunch

14h-14h45 Vittorino Pata (Politecnico Milano)

Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity

15h-15h45 discussion and closure

Abstracts

Alain Miranville: Mathematical models for brain lactate kinetics

Our aim in this talk is to discuss mathematical models
for the kinetics of lactactes in glial cells. We will also give
numerical simulations and comparisons with real data.

Alejandro Franco: White and Black Box computational approaches for the prediction of better batteries.

In this presentation I discuss our ongoing efforts on the development of computational approaches for the simulation of self-organization phenomena along the fabrication process of lithium ion battery electrodes. White Box approaches, based on Coarse Grained Molecular Dynamics, and Black Box approaches, based on machine learning algorithms, are synergistically used to predict the influence of fabrication parameters on the electrode electrochemical performance at the cell level. Theoretical principles, implementation aspects and experimental validation strategies of such tools are presented. Their practical implications towards the fabrication process optimization and the integration of new battery chemistries are discussed.

Vittorino Pata: Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity

We consider a Navier-Stokes-Voigt fluid model where the instantaneous
kinematic viscosity has been completely replaced by a memory term
incorporating hereditary effects, in presence of Ekman damping.
Unlike the classical Navier-Stokes-Voigt system, the energy balance involves the spatial
gradient of the past history of the velocity rather than providing an instantaneous control
on the high modes. In spite of this difficulty, we show that our system is dissipative in
the dynamical systems sense and even possesses regular global and exponential attractors
of finite fractal dimension. Such features of asymptotic well-posedness in absence of instantaneous
high modes dissipation appear to be unique within the realm of dynamical systems arising from fluid models (joint work with F. Di Plinio, A. Giorgini and R. Temam).

Filippo dell'Oro: Semidiscrete schemes for evolution equations with memory

We introduce a new mathematical framework for the time discretization of evolution equations with memory. As a model, we focus on an abstract version of the Gurtin-Pipkin equation modeling hereditary heat conduction. Well-posedness and exponential stability of the discrete scheme are shown, as well as the convergence to the solutions of the continuous problem when the time-step parameter vanishes.

Maurizio Grasselli: The Cahn-Hilliard equation and the separation property

Consider a binary system (e.g., an alloy) whose species have
concentrations c_1 and c_2 where c_1+c_2=1. If this system is cooled
down below a given critical temperature (which depends on the mixture)
then a phase separation process takes place. This pattern formation
phenomenon, a paradigm of phase segregation, can be described fairly
well by the so-called Cahn-Hilliard equation. This is a nonlinear
evolution equation for one of the concentrations (say, e.g., c=c_1).
More precisely, the Cahn-Hilliard equation can be seen as a conserved
gradient flow generated by the first variation of a (possibly
nonlocal) nonconvex free energy functional. The initial concentration
cannot be a pure phase, i.e., it cannot be identically equal to 0 or
1. However, it can be a pure phase on a set of positive measure. An
interesting question is the following: does the concentration become
(uniformly) mixed in finite time? This is the so-called (strict)
separation (from the pure phases) property. We can reformulate it as
follows: is there any constant r in (0,1), which only depends on a
bound on the total mass, such that c takes values in [r,1-r] from a
certain time on? If so, the entropy term in the free energy functional
becomes globally smooth. In particular, the validity of this property
entails a complete regularity theory for the Cahn-Hilliard equation
and a rather detailed picture of the longtime behavior of its
solutions. I intend to discuss the state-of-the-art of the above
issue.

Jean-Paul Chehab: Stabilization schemes for Phase field equations

We present here several approaches that allow to stabilize semi-implicit numerical schemes devoted to the simulation of reaction diffusion problem with a special focus on Phase Fields equations.
Both finite differences and finite element discretizations are proposed.

Benedetta Noris: Orbital stability of the ground states for the nonlinear Schrödinger equation with harmonic potential

We consider the nonlinear Schrödinger equation with harmonic potential in R^N and show that least energy standing wave solutions are always orbitally stable, as opposed to least action standing wave solutions which may be orbitally unstable. This is a work in collaboration with L. Jeanjean and D. Bonheure.

Monica Conti: A viscoelastic equation with nonlinear density and memory effects

We study a nonlinear viscoelastic equation suitable to modeling extensional vibrations of thin rods with nonlinear material density, depending on the velocity, and memory effects. We discuss the well-posedness of the model and the long-term behavior of the associated semigroup in the past-history framework. In particular, we show that the sole weak dissipation given by the memory term is enough to ensure existence and optimal regularity of the global attractor.

Cecilia Cavaterra: An inverse problem arising from cardiac electrophysiology.

We consider a model describing the evolution of the electric potential in the heart tissue. The goal is the determination of a small inhomogeneity inside the domain occupied by the heart from observations of the potential on the boundary. Such a problem is related to the detection of myocardial ischemic regions, characterized by severely reduced blood perfusion and consequent lack of electric conductivity. Both theoretical analysis and numerical reconstruction techniques are developed.