3rd AMARENA DaY

Amiens-Milano-poitiers REunion on Numerical and mathematical Analysis on phase fields models

April 27th 2015

3rd AMARENA DaY

Amiens-Milano-poitiers REunion on Numerical and mathematical Analysis on phase fields models

April 27th 2015

Laboratoire Amiénois de Mathématique

Fondamentale et Appliquée

CNRS UMR 7352

Contact: Jean-Paul Chehab or Alberto Farina

PHC Galileo

Program

8h30 Welcome in LAMFA

9h Conferences (Salle de séminaire du LAMFA)

Alain MIRANVILLE (Univ. Poitiers)

Some generalizations of the Cahn-Hilliard equation

Vittorino PATA (POLIMI, Milano)

A minimal approach to the theory of global attractors

Pierre Garnier (LAMFA, Amiens)

Long time behaviour of dispersive PDEs with generalized damping

12h Lunch

14h15 Conferences

Cecilia CAVATERRA (Univ. Studi Milano)

Non-isothermal viscous Cahn-Hilliard equation
with inertial term and dynamic boundary conditions

Maurizio GRASSELLI (POLIMI, Milano)

On a diffuse interface model for tumor growth

16h15-16h45 discussions and end of the meeting

abstracts

Maurizio GRASSELLI (POLIMI, Milano)

On a diffuse interface model for tumor growth

Hawkins-Daruud et al. ((2013) J. Math. Biol. 67 1457?1485) proposed a diffuse interface model for tumor growth. This model consists of the Cahn-Hilliard equation for the tumor cell fraction nonlinearly coupled with a reaction-diffusion equation for the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function. The system is equipped with no-flux boundary conditions which entail the conservation of the total mass, that is, the spatial average of the sum of the two fractions. I present some theoretical results I recently obtained jointly with Sergio Frigeri and Elisabetta Rocca (both at WIAS, Berlin). More precisely, these results are concerned with the well-posedness in a weak setting, the existence of a strong solution and the existence of a global attractor.

Vittorino PATA (POLIMI, Milano)

A minimal approach to the theory of global attractors

For a semigroup S(t) acting on a metric space X, we give a notion of global attractor based only on the minimality with respect to the attraction property. Such an attractor is shown to be invariant whenever S(t) is "asymptotically closed". As a byproduct, we generalize earlier results on the existence of global attractors in the classical sense. An application to a concrete model arising in the PDE context will be given.

Cecilia CAVATERRA (Univ. Studi Milano)

Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions

We consider a non-isothermal modified viscous
Cahn–Hilliard equation.
Such an equation is characterized by an inertial term
and it is coupled with a hyperbolic heat equation from
the Maxwell–Cattaneo’s law.
We analyze the case in which the order parameter is subject
to a dynamic boundary condition. This assumption requires
a more refined strategy to extend the previous results to
the present case. More precisely, we first prove the
well-posedness for solutions with finite energy as well as
for weak solutions. Then we establish the existence of a
global attractor. Finally, we prove the convergence of any
given weak solution to a single equilibrium by using a suitable
Lojasiewicz–Simon inequality.

Alain MIRANVILLE (Univ. Poitiers)

Some generalizations of the Cahn-Hilliard equation

Our aim in this talk is to discuss generalizations of the Cahn-Hilliard
equation. In particular, we will discuss applications in biology and image
inpainting.

Pierre GARNIER (Univ. Picardie)

Long time behaviour of dispersive PDEs with generalized
damping

We study the long-time behavior of of the solutions of the Benjamin-Bona-Mahony and the Korteweg-de Vries equations with damping acting on frequencies.