4th AMARENA DaY

Amiens-Milano-poitiers REunion on Numerical and mathematical Analysis

May 9th 2016

4th AMARENA DaY

Amiens-Milano-poitiers REunion on Numerical and mathematical Analysis

May 9th 2016

Laboratoire Amiénois de Mathématique

Fondamentale et Appliquée

CNRS UMR 7352

PICS CNRS AMaMMO

Program

8h30 Welcome in LAMFA

9h Conferences (BC101)

Vittorino Pata (Politecnico di Milano)

Viscoelasticity with time-dependent memory kernels

Alain Miranville (Université de Poitiers)

Cahn-Hilliard inpainting

Cecilia Cavaterra (Università degli studi di Milano)

Global strong solutions of the full Navier-Stokes

and Q-tensor system in 2D: existence and long-time behavior

Maurizio Grasselli (Politecnico di Milano)

Cahn-Hilliard-Navier-Stokes systems with generalized Navier boundary conditions

abstracts

Vittorino Pata (Politecnico di Milano)

Viscoelasticity with time-dependent memory kernels

We consider an integrodifferential equation arising in the theory of viscoelasticity characterized by a memory kernel explicitly depending on time, allowing for instance to describe the dynamics of aging materials. From the mathematical viewpoint, this translates into the study of dynamical systems acting on time-dependent spaces.

After giving the proper notion of solution, we discuss well-posedness results and the asymptotic behavior of the solutions.

Maurizio Grasselli (Politecnico di Milano)

Cahn-Hilliard-Navier-Stokes systems with generalized Navier boundary conditions

The motion of an isothermal mixture of two immiscible and incompressible fluids subject to phase separation can be described by the well-known model H. This is a diffuse interface model which consists of the Navier-Stokes equations for the average velocity which are subject to a force depending on the difference of the relative concentrations of the two fluids. The evolution of this concentration is governed by a convective Cahn-Hilliard equation. The system is also known as Cahn-Hilliard-Navier-Stokes (CHNS) system. In the existing literature there are many theoretical results on CHNS systems equipped with no-slip (or periodic) boundary conditions for the fluid velocity and no-flux (or periodic) boundary conditions for the concentration and the chemical potential. However, it has been observed that the moving contact line, defined as the intersection of the fluid-fluid interface with the solid wall, is incompatible with the no-slip boundary condition and its references). More precisely, there is a velocity discontinuity at the moving contact line, and the tangential force exerted by the fluids on the solid surface in the vicinity of the contact line becomes infinite. Thus, in immiscible two-phase flows, none of the mentioned boundary conditions can account for the moving contact line slip velocity profiles obtained from simulations.

Therefore, new boundary conditions are required to describe the observed phenomena. An example of such conditions are the so-called generalized Navier boundary conditions (GNBC). I intend to discuss the existence of a global weak solution to the CHNS system endowed with GNBC accounting for some boundary diffusion for the concentration.

I shall also show that any weak energy solution converges to a single equilibrium. Some open issues will be mentioned in the end.

Cecilia Cavaterra (Università degli studi di Milano)

Global strong solutions of the full Navier-Stokes and Q-tensor system in 2D: existence and long-time behavior

We consider a full Navier-Stokes and Q-tensor system for incompressible liquid crystal flows of nematic type.

In the two dimensional periodic case, we prove the existence and uniqueness of global strong solutions that are uniformly bounded in time. This result is obtained without any smallness assumption on the physical parameter ξ that measures the ratio between tumbling and aligning effects of a shear flow exerting over the liquid crystal directors.

Moreover, we show the uniqueness of asymptotic limit for each global strong solution as time goes to infinity and provide an uniform estimate on the convergence rate.

Alain Miranville (Université de Poitiers)

Cahn-Hilliard inpainting

Our aim in this talk is to discuss variants of the Cahn-Hilliard equation in view of applications to image inpainting. We will present theoretical results as well as numerical simulations.