Special Session 54: Nonlocal dynamics and complex patterns in phase-separation

A sixth-order Cahn--Hilliard equation for curvature effects in pattern formation

Pierluigi Colli University of Pavia Italy Co-Author(s):

Abstract: This talk is concerned with a sixth-order Cahn--Hilliard system, which represents a higher-order variant of the well-known Cahn--Hilliard equation. In the system, the evolution equation is complemented with a source term, where the control variable enters as a distributed mass regulator. The presence of further spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, for a more accurate description of isothermal phase separation dynamics in complex materials systems. The well-posedness and optimal control are discussed for the related initial and boundary value problem. Well-posedness is shown when assuming a smooth double-well potential as part of the free energy. Then, the optimal control problem is addressed: existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality involving the solution to the adjoint problem. These results have been obtained in a recent collaboration with G. Gilardi (University of Pavia), A. Signori (Polytechnic of Milan) and J. Sprekels (WIAS Berlin).

Stochastic diffuse interface models with conservative noise

Andrea Di Primio Politecnico di Milano Italy Co-Author(s):    Maurizio Grasselli, Luca Scarpa

Abstract: In this talk, we consider the Cahn--Hilliard and the conserved Allen--Cahn equations with logarithmic type potential and conservative noise in a periodic domain. These features ensure that the order parameter takes its values in the physical range and, albeit the stochastic nature of the problems, that the total mass is conserved almost surely in time. For the Cahn--Hilliard equation, existence and uniqueness of probabilistically-strong solutions is shown up to the three-dimensional case. For the conserved Allen--Cahn equation, under a restriction on the noise magnitude, existence of martingale solutions is proved even in dimension three, while existence and uniqueness of probabilistically-strong solutions holds in dimension one and two. The analysis is carried out by studying the Cahn--Hilliard/conserved Allen--Cahn equations jointly, that is a linear combination of both the equations, which has an independent interest.

Nonlocal to local convergence of the degenerate Cahn-Hilliard equation

Charles Elbar Sorbonne Universite France Co-Author(s):    Jakub Skrzeczkowski

Abstract: There has been recently an important interest in deriving rigorously the Cahn-Hilliard equation from the nonlocal equation. Since we are motivated by models for the biomechanics of living tissues, it is useful to include degenerate motilities. In this framework, we present a method to show the convergence of the nonlocal to the local degenerate Cahn-Hilliard equation. The method includes the use of nonlocal Poincare and compactness inequalities.

New results for the Cahn-Hilliard equation

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

Abstract: In this talk I will present some recent results concerning the well-posedness of the Cahn-Hilliard equation. This is a joint work with Monica Conti (Politecnico di Milano), Pietro Galimberti and Stefania Gatti (Universit\`{a} di Modena e Reggio Emilia).

Nonlocal Cahn-Hilliard-Darcy systems

Maurizio Grasselli Politecnico di Milano Italy Co-Author(s):    Cecilia Cavaterra, Sergio Frigeri

Abstract: A nonlocal Cahn-Hilliard-Darcy system for an incompressible mixture of two fluids consists of a convective nonlocal Cahn-Hilliard equation coupled with a Darcy`s law through the Korteweg force. We will discuss some recent results that have been proven for such system, focusing on the case where the mixing entropy density is of Boltzmann-Gibbs type, the kinematic viscosity depends on the order parameter, and the mobility is degenerate at the pure phases. The main issues will be the existence of different types of solutions, their regularization properties, and the longtime behavior of the associated dynamical system.

Nonlocal-to-local convergence rates for a Navier-Stokes-Cahn-Hilliard system

Patrik Knopf University of Regensburg Germany Co-Author(s):    Christoph Hurm, Andrea Poiatti

Abstract: We discuss the nonlocal-to-local convergence of strong solutions to a Navier-Stokes-Cahn-Hilliard model (Model H) with singular potential describing immiscible, viscous two-phase flows with matched densities. This means that we show that the strong solutions of the nonlocal Model H converge to the corresponding strong solution of the local Model H as the weight function in the nonlocal interaction kernel approaches the delta distribution. Compared to previous results in the literature, our main novelty is to further establish concrete rates for this nonlocal-to-local convergence.

Convergence of a nonlocal to a local phase field system with inertial term

Shunsuke Kurima Tokyo University of Science Japan Co-Author(s):    Pierluigi Colli, Shunsuke Kurima, Luca Scarpa

Abstract: There are some studies on local asymptotics for nonlocal problems. For example, Davoli--Scarpa--Trussardi (2021) and Abels--Terasawa (2022) have studied nonlocal-to-local convergence of Cahn--Hilliard equations. On the other hand, regarding phase field systems, in the case of a conserved phase field system related to entropy balance, nonlocal-to-local convergence has already been confirmed (K. (2022)). In this talk, we focus on convergence of a nonlocal phase field system with inertial term to a parabolic-hyperbolic phase field system. This is a joint work with Professors Pierluigi Colli (University of Pavia) and Luca Scarpa (Polytechnic University of Milan).

A Cahn-Hilliard-Darcy system with dynamic boundary conditions

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

Abstract: We will present some mathematical results for a Cahn-Hilliard-Darcy system complemented with dynamic boundary conditions of Cahn-Hilliard type: namely, we wiill assume that the trace of the bulk order parameter also satisfies a suitable fourth order evolutionary equation on the boundary. In particular we will prove existence of weak solutions and investigate the (asymptotic) relations linking together various types of dynamic boundary conditions. Moreover we will discuss the problem of the existence of the trace of the bulk velocity and its connection with the boundary velocity.

Long time behavior of the solution to a stochastic Allen-Cahn-Navier-Stokes system with logarithmic potential.

Margherita Zanella Politecnico di Milano Italy Co-Author(s):

Abstract: We consider a stochastic version of the Allen-Cahn-Navier-Stokes system in a smooth two-dimensional domain with random initial data. The system consists of a Navier-Stokes equation coupled with a convective Allen-Cahn equation, with two independent sources of randomness given by general multiplicative-type Wiener noises. In particular, the Allen-Cahn equation is characterized by a singular potential of logarithmic type as prescribed by the classical thermodynamical derivation of the model. We analyze the long-time behavior of the (probabilistically-strong unique) solution: we establish the existence, uniqueness and asymptotic stability of the invariant measure associated to the system. The talk is based on a joint work with A. Di Primio and L. Scarpa.