8th AMARENA DAYS
Amiens-Milano-poitiers REunion on Numerical and mathematical Analysis
12 - 13 May 2022
Salle de thèses, Ecole doctorale STS, UPJV

Program

## First day, May 12

9h00: Welcome
9h30-10h15: Vittorino Pata (Politecnico di Milano) Decay properties of the supercritical MGT-Fourier model
10h15-11h: Cecilia Cavaterra (Università di Milano, La Statale) Recent results on nonlocal Cahn-Hilliard-Hele-Shaw systems
11h-11h15: break
11h15 -12h: Alejandro A. Franco (Université de Picardie, LRCS, IUF) A digital twin for the optimization of lithium ion battery manufacturing processes
12h-14h: lunch
14h-14h45: Monica Conti (Politecnico di Milano) Some unexplored questions arising in linear viscoelasticity. Part I
15h-15h45: Filippo Dell'Oro (Politecnico di Milano) Some unexplored questions arising in linear viscoelasticity. Part II
15h45-16h: break
16h-16h45: Caterina Calgaro (Université de Lille, Laboratoire Paul Painlevé) A finite volume approximation of the temperature in a low Mach combustion model

## Second day, May 13

9h30-10h15: Alain Miranville (Université de Poitiers) Mathematical models for glial cells
10h15-11h: Youcef Mammeri (Université de Picardie, LAMFA) On the well-posedness of WBK equations
11h-11h15: break
11h15 -12h: Matthieu Brachet (Université de Poitiers) Discretization of Cahn-Hilliard equations
12h-14h: lunch
14h-14h45: Maurizio Grasselli (Politecnico di Milano) A phase-field model for RNA-protein dynamics
14h45-16h: discussion and closure
Abstracts

Vittorino Pata: Decay properties of the supercritical MGT-Fourier model

We address the energy transfer in the differential system  u_{t t t} + \alpha u_{t t} - \beta \Delta^m u_t - \gamma \Delta^m u = -\eta \Delta \theta; \theta_t - \kappa \Delta \theta =\eta \Delta u_{t t}+ \alpha\eta \Delta u_t  where m=1,2, made by a Moore-Gibson-Thompson equation in the supercritical regime, hence antidissipative, coupled with the classical heat equation. The two values of m correspond to two different physical models. The focus is the analysis of the energy transfer between the two equations, particularly when the first one stands in the supercritical regime, and exhibits an antidissipative character. The principal actor becomes then the coupling constant \eta, ruling the competition between the Fourier damping and the MGT antidamping. Indeed, we will show that a sufficiently large \eta is always able to stabilize the system exponentially fast. The presence of the Laplacian versus the Bilaplacian introduces some differences in the mathematical treatment. The results have been obtained in collaboration with M. Conti, F. Dell'Oro and L. Liverani.

Cecilia Cavaterra: Recent results on nonlocal Cahn-Hilliard-Hele-Shaw systems

We study a Cahn-Hilliard-Hele-Shaw system for an incompressible mixture of two fluids. The relative concentration difference is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential while the fluid velocity obeys a Darcy’s law depending on the Korteweg force. Similar systems, called in general Cahn-Hilliard-Darcy systems, are also used to describe fluid flow in a porous medium as well as to model solid tumor growth through diffuse interfaces. We present a series of results, namely, the existence of a global weak solution which satisfies an energy identity, the existence of a strong solution, further regularity properties, and uniqueness issues. Some work in progress on the same system with sources will also be mentioned.

Alejandro Franco: A digital twin for the optimization of lithium ion battery manufacturing processes

In this lecture I discuss an infrastructure for accelerated optimization of the manufacturing process of Lithium Ion Batteries (LIBs) we are developing within the context of the ARTISTIC project. Such infrastructure is supported on a hybrid approach encompassing experimental characterizations, a physics-based multiscale modeling workflow and machine learning models. Different steps along the LIB cells manufacturing process are simulated, such as the electrode slurry, coating, drying, calendering and electrolyte infiltration. The multiscale physical modeling workflow couples experimentally-validated Coarse Grained Molecular Dynamics, Discrete Element Method and Lattice Boltzmann simulations and it allows predicting the impact of the process parameters on the final electrode mesostructure in three dimensions. The predicted electrode mesostructures are injected in a Finite Element Method-supported performance simulator capturing the influence of the pore networks and spatial location of carbon-binder within the electrodes on their electrochemical behavior. Machine learning models are used to accelerate the physical models’ parameterization, to derive surrogate models and to unravel manufacturing parameters interdependencies from the physical models’ predictions and experimental data. Furthermore, Bayesian Optimization is used for inverse design of manufacturing processes. The predictive capabilities of this digital twin are illustrated with results for different electrode formulations. Finally, the free online battery manufacturing simulation services offered by the project3 to optimize battery electrodes are illustrated through several examples.

Monica Conti: Some unexplored questions arising in linear viscoelasticity. Part I
Filippo Dell'Oro: Some unexplored questions arising in linear viscoelasticity. Part II

We consider an abstract integrodifferential equation modeling the dynamics of linearly viscoelastic solids. The equation is known to generate a semigroup on a certain phase space, whose asymptotic properties have been the object of extensive studies in the last decades. Nevertheless, some relevant questions still remain open, with particular reference to the decay rate of the semigroup compared to the decay of the memory kernel, and to the structure of the spectrum of the infinitesimal generator of the semigroup. In these talks we intend to provide some answers.

Caterina Calgaro : A finite volume approximation of the temperature in a low Mach combustion model

Given a specific low-Mach model expressed in velocity, pressure and temperature variables, we focus our attention on the convergence of a finite volume method to approximate the solution of a convection-diffusion equation involving a Joule term. In particular: we establish a discrete version of a Gagliardo-Nirenberg inequality, to apply it to the analysis of the numerical scheme;we consider a discretization of the Joule term consistent with the non linear diffusion one, in order to ensure the maximum principle on the solution; we prove the convergence of the numerical scheme by compactness arguments.

Alain Miranville : Mathematical models for glial cells

Our aim in this talk is to discuss mathematical models for lactate concentration in glial cells. Lactate is an essential marker for brain cancers (glioma). In particular, we discuss a Cahn-Hilliard type model.

Youcef Mammeri : On the well-posedness of WBK equations

The Whitham-Broer-Kaup equations have been introduced by Kupershmidt in 1985 as a combination of three independent systems derived by the named authors. These systems describes water waves in shallow. Despite its physical similarity with the Boussinesq systems, its mathematical properties differ. Because the WBK is known to be linearly ill-posed, few studies are concerned with its theoretical study. I will show, with some assumptions on the parameters, that the system is actually locally well-posed (and globally for some nonlinearties). It is a joint work with Nabil Bedjaoui and Rajesh Kumar.

Matthieu Brachet : Discretization of Cahn-Hilliard equations

Solutions of Cahn-Hilliard equations satisfy some properties such as energy decreasing, conservation of mass or convergence to equilibrium. After discretization, some properties may be lost or weakened. We analyze the properties of numerical schemes and consider some open questions. The talk will be illustrated by numerical simulations.

Maurizio Grasselli : A phase-field model for RNA-protein dynamics

We consider a phase-field model which accounts for the formation of protein-RNA complexes subject to phase segregation. The two complexes obey a system of two Cahn-Hilliard equations with Flory–Huggins potentials and reaction terms. This system is coupled with three reaction-diffusion equations for a protein and two RNA species. No-flux boundary conditions and suitable initial conditions are given. The main peculiarity is that the complexes do not exist at the initial time. Therefore the Cahn-Hilliard equations have zero initial conditions. Nonetheless, the existence of a global weak solution can be proven thanks to a new Lp(L2)-estimate of the chemical potential. Also, in dimension two, regularization in finite time and strict separation from pure phases can be established. This is a joint work with Luca Scarpa (Politecnico di Milano) e Andrea Signori (Università degli Studi di Pavia).