Special Session 3: Analysis of diffuse and sharp interface models

On a nonisothermal tumor growth model of Caginalp type

Giulia Cavalleri WIAS Germany Co-Author(s):    Pierluigi Colli, Elisabetta Rocca

Abstract: We study a nonisothermal phase-field system of Caginalp type that describes tumor growth under thermal therapy. The model couples a possibly viscous Cahn--Hilliard equation, governing the evolution of the healthy and tumor phases, with an equation for the heat balance, and a reaction-diffusion equation for the nutrient concentration. The resulting nonlinear system incorporates chemotaxis and active transport effects, and hyperthermia appears as a control variable. First, we prove well-posedness for the initial-boundary value problem and additional regularity. Then, we define a suitable cost functional and show the existence of optimal controls. Finally, we analyze the differentiability of the control-to-state operator and establish necessary first-order conditions of optimality. These results have been obtained in collaboration with Pierluigi Colli (University of Pavia) and Elisabetta Rocca (University of Pavia).

On the Cahn-Hilliard equation with nonlinear diffusion: the non-convex case

Monica Conti Politecnico di Milano Italy Co-Author(s):    Stefania Gatti, Andrea Giorgini and Giulio Schimperna

Abstract: We investigate the Cahn-Hilliard equation with nonlinear diffusion and non-degenerate mobility modeling phase separation phenomena in complex systems (e.g., crystals and polymers). Previous results in the literature on this model relied on the strong convexity assumption of the gradient part of the energy, which excludes relevant cases. In this work, we remove the convexity condition and establish new qualitative properties of solutions under general assumptions on the diffusion and mobility functions.

Deterministic particle approximation of a fourth-order PDE

Charles Elbar Universite Claude Bernard Lyon 1 France Co-Author(s):    Alejandro Fernandez-Jimenez

Abstract: Many partial differential equations describe how a macroscopic density of particles/cells evolves in time. So, it is a natural question to ask whether there is a microscopic (deterministic) model that leads to a given macroscopic PDE. For second-order aggregation-diffusion equations, this connection has been proved, starting with the porous medium equation in 2001. In this work, we extend that framework to a fourth-order equation, inspired by applications in cell-cell adhesion in biological systems. This is a joint work with Alejandro Fernandez-Jimenez.

Asymptotic analysis of Robin type transmission problems for gap junctions

Takeshi Fukao Ryukoku University Japan Co-Author(s):    Takeshi Fukao

Abstract: In this talk, we focus on Robin type transmission conditions that describe gap junctions arising in biology, and we study the well-posedness of transmission problems of Allen--Cahn type as well as the asymptotic analysis with respect to the permeability parameter. For parabolic equations, transmission problems with Dirichlet--Neumann type conditions have been widely studied. Here, we treat Robin type transmission conditions instead. A distinctive feature of these conditions is that they allow discontinuity in the values of the unknown functions on the interface shared by the two domains. This can be well understood by recalling that, in the Dirichlet--Neumann case, the trace condition ensures continuity of values across the interface. The Robin type condition naturally expresses the degree of interaction on the interface in terms of the permeability. We first establish the well-posedness of the Robin type problem as a prerequisite. For this purpose, the abstract theory of evolution equations governed by subdifferential operators is applicable. We then discuss the limiting procedures when the permeability converges to zero and when it diverges to infinity. It can be shown that the former limit yields a split system consisting of separate Allen--Cahn equations on each domain, while the latter limit leads to a transmission problem of Dirichlet--Neumann type, both results being consistent with the physical meaning of the permeability. Furthermore, we address the case where the permeability depends on time, as well as the case where the permeability diverges to infinity at a finite time, showing that the system can be extended beyond that instant.

A Generalized Cahn-Hilliard equation with non-degenerate mobility: Well Posedness and Convergence to the Classical Cahn-Hilliard

Pietro Galimberti University of Ferrara Italy Co-Author(s):    Monica Conti, Pietro Galimberti, Stefania Gatti and Andrea Giorgini

Abstract: We study a generalized Cahn--Hilliard equation, based on an unconstrained theory proposed by Duda, Sarmiento and Fried in 2021, with non-degenerate mobility and nonlinear terms of logarithmic type. We prove well posedness of weak solutions, propagation of regularity and a type of separation from the pure states property for weak solutions, also in three space dimensions. Moreover, given that this model can be interpreted as a perturbation of the classical Cahn-Hilliard, we prove convergence of weak solutions to weak solutions of the Cahn-Hilliard equation on finite time intervals.

Diffuse interface models on evolving surfaces: Modeling and analysis

Harald Garcke Department for Mathematics Germany Co-Author(s):

Abstract: We consider geometric evolution problems consisting of evolution equations for a closed hypersurface coupled to parabolic equations on this evolving surface. More precisely, the evolution of the hypersurface is determined by a geometric flow that depends on a quantity defined on the surface via a diffusion equation. This system arises as a gradient flow of an energy functional. Assuming suitable parabolicity conditions, we derive short-time existence for the system. The proof is based on a linearization and contraction argument. Afterwards, several properties of the solution are analyzed. In particular, we emphasize to what extent the surface in our setting evolves in the same way as under the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but we give an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. We will also discuss how Willmore type flow can be coupled to the Cahn-Hilliard equation on an evolving surface. The resulting system is a coupled Cahn-Hilliard/Canham-Helfrich flow. Finally, we construct an embedded hypersurface that develops a self-intersection over time and we discuss how solutions can be computed numerically with the help of an evolving surface finite element discretization.

Recent advances on the 2D Cahn-Hilliard equation with nondegenerate mobility

Stefania Gatti Universit\` degli Studi di Modena e Reggio Emilia Italy Co-Author(s):    Monica Conti, Pietro Galimberti, Andrea Giorgini

Abstract: In this talk, we will discuss some recent results on the 2D Cahn-Hilliard equation with nondegenerate concentration-dependent mobility and logarithmic potential. More precisely, we proved that any weak solution is unique, exhibits propagation of uniform-in-time regularity, and stabilizes towards an equilibrium state of the Ginzburg-Landau free energy for large times. These results improve the state of the art dating back to a work by Barrett and Blowey and were obtained in a joint work with Monica Conti, Pietro Galimberti and Andrea Giorgini.

Estimates on free-boundary propagation for solutions to stochastic porous-media equations

Guenther Gruen Department of Mathematics, University of Erlangen-Nuremberg Germany Co-Author(s):    Joshua Utley

Abstract: We study the impact of noise on the expected propagation in time of the boundary of the spatial support of solutions to stochastic porous-media equations with non-linear conservative noise, i.e. multiplicative noise inside a convective term. Starting from a recent result together with M. Sauerbrey on finite speed of propagation for kinetic solutions of such equations (for which existence and non-negativity is known due to previous work of B. Fehrman and B. Gess), we formulate upper estimates on propagation rates for small times. With respect to scaling, these estimates coincide with the estimates known to be optimal in the deterministic setting. For large times, however, we observe a scaling transition in the stochastic case which gives strong indication that the presence of conservative noise terms enhances spreading significantly. For the proof, we rely on integral estimates and appropriate filtering techniques. This is a joint work with Joshua Utley, Friedrich-Alexander University Erlangen - Nuremberg.

Nonlocal-to-Local Convergence of the Cahn--Hilliard Equation and its Operator

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

Abstract: We prove convergence of a sequence of weak solutions to the nonlocal Cahn-Hilliard equation to the weak solution to the corresponding local Cahn-Hilliard equation. The analysis is done in the case of sufficiently smooth bounded domains with Neumann boundary condition and a $W^{1,1}$-kernel. The proof is based on an energy method. Additionally, we prove the strong $L^p$-convergence of the nonlocal operator to a local differential operator together with a rate of convergence. The analysis also includes more singular kernels.

Two-phase flows with bulk-surface interaction in an evolving domain

Patrik Knopf University of Regensburg Germany Co-Author(s):    Yadong Liu

Abstract: We discuss a thermodynamically consistent model, which describes the time evolution of a two-phase flow in an evolving domain. The movement of the free boundary of the domain is driven by the velocity field of the mixture in the bulk, which is determined by a Navier--Stokes equation. In order to take interactions between bulk and boundary into account, we further consider two materials on the boundary, which may be the same or different materials as those in the bulk. The bulk and the surface materials are represented by respective phase-fields, whose time evolution is described by a bulk-surface convective Cahn--Hilliard equation. This approach allows for a transfer of material between bulk and surface as well as variable contact angles between the diffuse interface in the bulk and the boundary of the domain. To provide a more accurate description of the corresponding contact line motion, we include a generalized Navier slip boundary condition on the velocity field. Based on local mass balance laws, we derive our model from scratch in two different ways: by the Lagrange Multiplier Approach and (in the case of matched densities and no mass flux between bulk and surface) by the Energetic Variational Approach. We further show that our model generalizes previous models from the literature, which can be recovered from our system by either dropping the dynamic boundary conditions or assuming a static boundary of the domain.

Finite-dimensional attractors to 2D NSE-Allen-Cahn equations with irregular potentials

Dalibor Prazak Charles University, Faculty of Mathematics and Physics Czech Rep Co-Author(s):

Abstract: We consider 2D Navier-Stokes system coupled to Allen-Cahn equation with irregular potentials. This leads to a differential inclusion with maximal monotone term. Existence of finite-dimensional attractor is shown using the method of l-trajectories, where the key step is an L^1-estimate of the difference of discontinuous terms.

Velocity control in Navier-Stokes-Cahn-Hilliard models with curvature effects

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

Abstract: In this talk, I discuss a class of coupled partial differential equation systems describing the interaction between an incompressible viscous fluid and a deformable membrane modeled through a phase-field variable. The analysis is carried out within the Navier-Stokes framework, allowing for the description of physically relevant regimes arising in biological membranes and amphiphilic materials. Particular attention is devoted to an associated optimal control problem in two spatial dimensions. The results highlight the interplay between fluid flow and membrane curvature in driving phase evolution and shaping membrane morphology. This work is part of a joint project with Prof. Hao Wu (Fudan University, Shanghai).

Two-phase flows with bulk-surface interaction: A Navier--Stokes--Cahn--Hilliard model with dynamic boundary conditions

Jonas Stange University of Regensburg Germany Co-Author(s):    Patrik Knopf

Abstract: We present a new diffuse interface model for incompressible, viscous fluid mixtures with bulk-surface interaction. This system consists of a Navier--Stokes--Cahn--Hilliard model in the bulk that is coupled to a surface Navier--Stokes--Cahn--Hilliard model on the boundary. Compared with previous models in the literature, the inclusion of an additional surface Navier--Stokes equation is motivated, for example, by biological applications. We prove the existence of weak solutions by means of a semi-Galerkin scheme combined with a fixed-point argument. To discretize the Navier--Stokes subsystem, we analyze a novel bulk-surface Stokes system and its corresponding bulk-surface Stokes operator, whose eigenfunctions serve as a natural basis to approximate the velocity fields. This is joint work with Patrik Knopf (University of Regensburg)

Structure and Stability of Global Attractors for a Cahn-Hilliard Tumor Growth Model

Sema Yayla Hacettepe University Turkey Co-Author(s):

Abstract: In this talk, we study the long-time dynamics of a Cahn-Hilliard tumor growth model with chemotaxis, focusing on the structure and stability of its global attractors. Using a \L{}ojasiewicz-Simon type inequality, we analyze the asymptotic behavior of trajectories and show that solutions converge to stationary states. This allows us to describe the geometric structure of the global attractor in terms of stationary solutions. We also investigate the stability of the attractor with respect to perturbations of the chemotaxis parameter and establish continuity properties of the attractors under suitable assumptions.

Global solutions of nematic liquid crystal flow

Yong Yu The Chinese University of Hong Kong Hong Kong Co-Author(s):    Yuan Chen, Soojum Kim, Shun Li, Pei Lyu

Abstract: Ericksen-Leslie hydrodynamical system describes the motion of nematic liquid crystal molecules. Lin-Liu model is a simplified Ericksen-Leslie model which preserves the main structure of the original Ericksen-Leslie system. In this talk, I survey several results concerning about the global weak solutions to the Lin-Liu model, and the long-time asymptotic behaviors of these solutions. The problems to be addressed will be on both 2D and 3D, and on both domains with fixed boundary and free boundary. All the results presented in the talk are supported by RGC grants of Hong Kong.