Special Session 10: Analysis of diffuse and sharp interface models

Phase field models for tumour growth incorporating finite visco-elasticity

Abramo Agosti University of Pavia Italy Co-Author(s):

Abstract: In this talk I will present phase field models for tumor growth, based on the Cahn-Hilliard equation, which incorporate the visco-elastic feedback of the surrounding tissues, which is characterized by nonlinear hyperelastic energy densities describing the mechanical behavior of soft tissues and coupled with the phase field variable. I will present recent analytical results for systems of finite visco-elasticity alone and for the fully coupled systems of PDEs. I will also present numerical simulations on patient-specific test cases, providing evidences of the importance to consider the elastic feedback of tissues to better forecast the tumour evolution.

Phase-Field Approaches for Shape Reconstruction in Elastic Inverse Problems

Andrea Aspri University of Milan Italy Co-Author(s):    Elena Beretta, Cecilia Cavaterra, Elisabetta Rocca, Marco Verani

Abstract: In this talk, I will discuss recent advances in solving elastic inverse problems, specifically focusing on the shape reconstruction of cavities and inclusions in a bounded linear isotropic medium using boundary measurements. Our approach leverages optimal control theory, reformulating the inverse problem as a minimization process. The objective is to minimize a misfit boundary functional or an energy-type functional, within the class of Lipschitz domains. To enhance the accuracy of the reconstruction, we introduce a regularization term that penalizes the perimeter of the cavity or inclusion being reconstructed. The optimization problem is tackled using a phase-field method, where the perimeter functional is approximated via the Modica-Mortola relaxation.

Existence and continuity of inertial manifolds for singularly perturbed conserved phase-field systems

Ahmed Bonfoh KFUPM Saudi Arabia Co-Author(s):

Abstract: We will prove that the solution semigroup generated by the singularly perturbed conserved phase-field system admits an inertial manifold of dimension independent of the perturbation parameter (the heat capacity). Moreover, we will show the convergence of the intersection of the inertial manifold with an absorbing set as the heat capacity tends to zero. This work was recently proven by the speaker in [Evol. Equ. and Control Theory, 11 no.4 (2022),1399-1454].

An optimal distributed control problem for a Cahn-Hilliard-Darcy system

Cecilia Cavaterra University of Milan Italy Co-Author(s):    M. Abatangelo, M. Grasselli, H. Wu

Abstract: We consider a Cahn-Hilliard-Darcy system with mass sources, equipped with an impermeability condition for the (volume) averaged velocity as well as homogeneous Neumann boundary conditions for the phase field and the chemical potential. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought, for instance, as a drug or a nutrient in applications to solid tumor growth evolution. We present some recent results obtained in collaboration with M. Abatangelo, M. Grasselli, and H. Wu on a distributed optimal control problem in the two dimensional setting with a cost functional of tracking-type. These results have been achieved In the physically relevant case, that is, assuming unmatched viscosities for the binary fluid mixtures and considering a Flory-Huggins type potential. In particular, we show that a second-order sufficient condition for the strict local optimality can also be proven.

A Cahn-Hilliard-Navier-Stokes model for tumor growth

Charles Elbar Sorbonne Universite France Co-Author(s):    Alexandre Poulain

Abstract: I will discuss a compressible Navier-Stokes Cahn-Hilliard model. The model, intended to describe tumor growth takes into possible non-matching densities and contrasts in mechanical properties (viscosity, friction) between the two phases of the fluid. It also comprises a term to account for possible exchange of mass between the two phases. I will give an idea of the scheme to prove the existence of weak solutions. Also, I will show a structure preserving numerical scheme and present some numerical simulations validating the properties of the numerical scheme and the behavior of the model.

Droplet models with singular potentials: equilibria and travelling waves

Lorenzo Giacomelli Sapienza University of Rome Italy Co-Author(s):

Abstract: We look at spreading phenomena under the action of singular potentials modeling repulsion between the liquid/gas interface and the substrate. First we briefly review the statics: depending on the form of the potential, the macroscopic profile of minimizers (when they exist) can be either droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient. Then we focus on the dynamics, assuming null slippage at the contact line. Based on formal arguments and numerical evidences, we report that travelling-wave solutions generically exist and have finite rate of bulk dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with steady states, travelling-wave solutions have finite energy for mild singularities. Time permitting, we also discuss a selection criterion for travelling waves, based on thermodynamically consistent contact-line conditions modeling friction at the contact line. Based on joint works with Riccardo Durastanti (Universiy of Naples Federico II).

On a Navier-Stokes-Cahn-Hilliard system with chemotaxis, active transport and reaction

Jingning He Hangzhou Normal University Peoples Rep of China Co-Author(s):    Hao Wu

Abstract: In this talk, we discuss a Navier--Stokes--Cahn--Hilliard model for viscous incompressible two-phase flows where the mechanisms of chemotaxis, active transport and reaction are taken into account. The evolution system couples the Navier--Stokes equations for the volume-averaged fluid velocity, a convective Cahn--Hilliard equation for the phase-field variable, and an advection-diffusion equation for the density of certain chemical substance. This system is thermodynamically consistent and generalizes the well-known ``Model H`` for viscous incompressible binary fluids. For the initial-boundary value problem with a physically relevant singular potential in a three dimensional bounded smooth domain, we first prove the existence and uniqueness of a local strong solution. When the initial velocity is small and the initial phase-field function as well as the initial chemical density are small perturbations of a local minimizer of the free energy, we establish the existence of a unique global strong solution. Afterwards, we show the uniqueness of asymptotic limit for any global strong solution as time goes to infinity and provide an estimate on the convergence rate.

Two-phase flows through porous media: A Cahn-Hilliard-Brinkman model with dynamic boundary conditions

Patrik Knopf University of Regensburg Germany Co-Author(s):    Pierluigi Colli, Andrea Signori, Giulio Schimperna

Abstract: We consider a new diffuse-interface model that describes creeping two-phase flows (i.e., flows exhibiting a low Reynolds number), especially flows that permeate a porous medium. The system of equations consists of a Brinkman equation for the volume averaged velocity field as well as a convective Cahn-Hilliard equation with dynamic boundary conditions for the phase-field, which describes the location of the two fluids within the domain. The dynamic boundary conditions are incorporated to model the interaction of the fluids with the wall of the container more precisely. In particular, they allow for a dynamic evolution of the contact angle between the interface separating the fluids and the boundary, and also for a convection-induced motion of the corresponding contact line. In this talk, modeling aspects as well as analytical results will be discussed.

Navier-Stokes equations with dynamic boundary conditions and related problems

Dalibor Prazak Charles University, Prague Czech Rep Co-Author(s):    B. Priyasad, M. Zelina

Abstract: We consider the evolutionary Stokes system subject to the so-called dynamic boundary condition \[ \beta \partial_t u + \alpha u + \nu[(Du)n]_\tau = h \qquad \textrm{on } \partial \Omega \] where $Du$ is the symmetric velocity gradient, $n$ is outer normal, and subscript $\tau$ denotes the tangential projection relative to $\partial \Omega$. \par Our first aim is to establish the basic $L^p$ theory, including the existence of an analytic semigroup and optimal $W^{k,p}$ estimates for $k=1$ and $2$. \par These results are then applied to related nonlinear systems: Navier-Stokes and Cahn-Hilliard Navier-Stokes equations.

The random separation property for stochastic phase-field models

Luca Scarpa Politecnico di Milano Italy Co-Author(s):    Federico Bertacco, Carlo Orrieri

Abstract: We introduce the concept of random separation property for stochastic phase-field models with singular potential. This consists in showing that almost every trajectory of the stochastic flow, with probability one, is detached from the potential barriers: the separation threshold depends on the trajectory itself and identifies thus a random variable. We illustrate the idea of the proof in the case of the stochastic Allen-Cahn equation, as well as qualitative properties of the random separation layer. Eventually, possible developments on the stochastic Cahn-Hilliard equation are also discussed. The works presented in the talk are based on joint collaborations with Federico Bertacco (Imperial College London) and Carlo Orrieri (University of Pavia).

On a Cahn-Hilliard-Brinkman-chemotaxis model with nonlinear sensitivity

Giulio Schimperna University of Pavia Italy Co-Author(s):

Abstract: We will present some mathematical results for a new model coupling the Cahn-Hilllard-Brinkman system with an evolutionary equation describing the active (chemotactic) transport of a chemical species influencing the phase separation process. Specifically, the model may arise in connection with tumor growth processes; mathematically speaking, it may be interesting in itself as it provides a new coupling between a Keller-Segel-like relation (the equation describing the evolution of the concentration of the chemical substance) and a fourth order (rather than a second order as in most models for chemotaxis) evolutionary system. Our main result will be devoted to proving existence of weak solution in the case when the chemotaxis sensitivity function has a controlled growth at infinity; a particular emphasis will be given to discussing the occurrence of critical exponents and to presenting a regularization scheme compatible with the a-priori estimates.

Active droplet formation in Cahn-Hilliard models with chemical reactions

Andrea Signori Politecnico di Milano Italy Co-Author(s):

Abstract: This presentation examines a Cahn-Hilliard model incorporating chemical reactions that facilitate active droplet formation while suppressing Ostwald ripening. We establish the model`s well-posedness and connect the diffuse interface model to a sharp interface model of Mullins-Sekerka type using matched asymptotic expansions. Our stability analysis demonstrates the coexistence of stable and unstable stationary solutions under specific conditions. Additionally, we utilize finite element computations to validate our results and illustrate various splitting scenarios.

The convergence of a nonlocal to a local anisotropic Cahn-Hilliard equation

Yutaka Terasawa Nagoya University Japan Co-Author(s):    Helmut Abels

Abstract: We consider a nonlocal Cahn-Hilliard equation with singular or regular kernel, where the kernel is symmetric and non-radial. We get the existence of a weak solution of the equation using the Moreau regularization of the potential and an implicit time discretization. We then prove convergence of suitable sequence of weak solutions of the equation to weak solutions of a local anisotropic Cahn-Hilliard equation. This talk is based on a joint work with Professor Helmut Abels (Regensburg Univ.).

Variational approach to pure traction and Signorini problem between linear and finite elasticity

Franco TOMARELLI Politecnico di Milano Italy Co-Author(s):    Francesco Maddalena (Politecnico di Bari) and Danilo Percivale (Universita` degli Studi di Genova)

Abstract: An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfills an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.