Special Session 85: Phase-field models and their singular limits

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.

Sharp interface limit of the 3D Navier-Stokes/Allen-Cahn system

Mingwen Fei Anhui Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk we shall prove convergence of the solutions of the 3D Navier-Stokes/Allen-Cahn system to solutions of the sharp interface model. Such limits are known as sharp interface limits. We generalize our previous two-dimensional results to the three-dimensional setting. This talk is based on a joint work with Helmut Abels, Yadong Liu and Maximilian Moser.

A Cahn-Hilliard-Darcy-Forchheimer surfactant model

Maurizio Grasselli Politecnico di Milano Italy Co-Author(s):    B. Ouyang, A. Poiatti, H. Wu

Abstract: We present a mathematical model for two-phase flows with surfactants in porous media, combining a Darcy--Forchheimer equation with two Cahn--Hilliard equations involving singular potentials. We show the existence of global weak solutions and that, over time, every solution converges to a single equilibrium. A key point is that this convergence holds at the level of weak solutions, without requiring extra regularity, which distinguishes this result from much of the existing literature. Further related results will also be discussed if time permits.

A weak-strong uniqueness principle for Mullins-Sekerka flow

Sebastian Hensel University of Leipzig Germany Co-Author(s):    Julian Fischer, Tim Laux, Theresa M. Simon

Abstract: We establish a weak-strong uniqueness principle for the two-phase Mullins-Sekerka equation in ambient dimension $d = 2$ and $3$: As long as a classical solution to the evolution problem exists, any weak De Giorgi type varifold solution (see for this notion the recent work with Stinson, Arch. Ration. Mech. Anal. 248, 8, 2024) must coincide with it. In particular, in the absence of topology changes such weak solutions do not introduce a mechanism for (unphysical) non-uniqueness. We also derive a stability estimate with respect to changes in the data. I will explain our method which is based on the notion of relative entropies for interface evolution problems, a reduction argument to a perturbative setting, and a stability analysis in this perturbative regime relying crucially on the gradient flow structure of the Mullins-Sekerka equation.

Two-phase flow with micropolar effects: Modelling, analysis and asymptotic limits

Kei Fong Lam Hong Kong Baptist University Hong Kong Co-Author(s):    Kin Shing Chan, Michael Eden, Baoli Hao, Bjorn Stinner

Abstract: Micropolar fluids are among the simplest cases of fluids with microstructures, where each fluid particle has its own internal rotations. Examples include ferrofluids, blood flows, bubbly liquids and liquid crystals. By combining the seminal work of A. Cemal Eringen and coworkers, with the diffuse interface approach for multiphase fluid flow, we develop new diffuse interface models for binary mixtures of micropolar fluids. Using recent advances in the mathematical analysis of such types of models, we establish existence of weak solutions and discuss several asymptotic limits, such as micropolar-to-nonpolar, nonlocal-to-local convergence, as well as flow within Hele-Shaw type thin domains.

Sharp Interface Limit for 3D Navier-Stokes/Allen Cahn Systems

Yadong Liu Nanjing Normal University Peoples Rep of China Co-Author(s):    Helmut Abels, Mingwen Fei, Yadong Liu, and Maximillian Moser

Abstract: In this talk, I will report a recent work on the sharp interface limit of a coupled Navier--Stokes/Allen--Cahn system in a three dimensional, bounded and smooth domain, when a parameter $\varepsilon > 0$ that is proportional to the thickness of the diffuse interface tends to zero, rigorously. The argument is based on the method of rigorous matched asymptotic expansions. In particular, we obtain optimal estimates for the linearized error system in $L^2$-Sobolev type spaces, which are second order in space of $\mathbf{w}$ and third order in space for $u$, with suitable weights. This provides better discription of the system close to the interface, which leads us to extend previous results from two dimensions to three dimensional case.

The Verigin problem with phase transition as Wasserstein flow

Alice Marveggio University of Bonn Germany Co-Author(s):    Anna Kubin and Tim Laux

Abstract: We study the modeling of a compressible two-phase flow in a porous medium. The governing PDE system is known as the Verigin problem with phase transition, which is the compressible analog to the Muskat problem. We prove the convergence of an implicit time discretization scheme using the Wasserstein distance, obtaining distributional solutions in the limit that satisfy an optimal energy-dissipation rate. Finally, we propose a phase-field model for the Verigin problem with phase transition and study its sharp-interface limit.

Singular limit of phase field models on varying surfaces

Matthias R\\"{o}ger Departement of Mathematics, TU Dortmund University Germany Co-Author(s):    Heiner Olbermann, UC Louvain and Benjamin Lledos

Abstract: We consider the sharp interface limit of phase field models on varying surfaces. Our approach uses the concept of BV functions over rectifiable currents introduced by Anzellotti, Delladio and Scianna [Annali di Matematica Pura ed Applicata, 1996]. In particular we will address Cahn-Hilliard type energies and a diffuse approximation of a J\{u}licher-Lipowsky type variational model for phase separated biomembranes. (This is joint work with Heiner Olbermann, UC Louvain and Benjamin Lledos, Universit\`{e} de N\^{i}mes)

Active Phase Separation and Self-Organizing Droplets

Andrea Signori Politecnico di Milano Italy Co-Author(s):    Harald Garcke, Kei Fong Lam, Robert N\urnberg

Abstract: Active phase separation leads to behaviors that significantly differ from those of classical systems. Instead of the usual coarsening, where larger droplets grow at the expense of smaller ones, chemically active mixtures can maintain stable populations of finite-sized droplets, which may even grow, divide, or form more complex structures. These dynamics have been proposed as simple models for protocells and early self-organizing biological structures. In this talk, I present a mathematical framework for active droplet dynamics based on two complementary descriptions: the phase-field Cahn--Hilliard equation and its sharp-interface limit given by the Mullins--Sekerka free-boundary problem. I discuss the relation between these models, address questions of well-posedness and stability in symmetric settings, and conclude with numerical simulations illustrating phenomena such as droplet division and shell formation.

Convergence Rates for Adversarial Training

Kerrek Stinson University of Utah USA Co-Author(s):    L. Bungert, R. Morris, R. Murray

Abstract: We discuss some qualitative and quantitative results analyzing adversarial training as the adversarial budget vanishes. First, with Bungert, we find that minimizers of the adversarial training problem converge in $L^1$ to a Bayes classifier that has minimal weighted perimeter, showing that adversarial training acts as a selection mechanism for the standard classification problem. Subsequent work by Morris and Murray showed that adversarial minimizers converge to Bayes classifiers in the (much stronger) Hausdorff metric at a rate that degrades with the dimension. Joining forces, we recover the generically optimal rate of convergence and prove that the Hausdorff distance between the adversarial minimizer and the Bayes classifier is $O(\epsilon )$ regardless of the ambient dimension.

Brakke`s inequality and the existence of Brakke flow for volume preserving mean curvature flow

Keisuke Takasao Kyoto University Japan Co-Author(s):    Andrea Chiesa

Abstract: We consider the existence of the weak solutions to the volume preserving mean curvature flow. The Brakke flow defined using Brakke's inequality is well known as one of the weak solutions to the mean curvature flow. On the other hand, the volume preserving mean curvature flow has been studied via $L^2$-flow solution, BV solution, and flat flow, but the corresponding Brakke flow had not been considered so far. In this talk, we define the suitable Brakke flow for the volume preserving flow and show its global existence. This talk is based on joint works with Andrea Chiesa.

Existence and Nonlocal-to-Local Convergence for Singular, Anisotropic Nonlocal Cahn-Hilliard Equations

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

Abstract: We study the nonlocal-to-local convergence for a nonlocal Cahn-Hilliard equation with anisotropic and singular kernels. In particular, we show convergence of weak solutions of the nonlocal Cahn-Hilliard equation to weak solutions of a corresponding anisotropic Cahn-Hilliard equation for suitable subsequences. Moreover, we show existence of weak solutions for the nonlocal equation under a condition, which guarantees existence of weak solutions for suitably localized or singular kernels. We put a focus on the existence part in the talk. This is based on a joint work with Helmut Abels (Regensburg Univ.)

Some problems for two-phase flow coupled with the Allen-Cahn equation

Yoshihiro Tonegawa Institute of Science Tokyo Japan Co-Author(s):

Abstract: I consider a simple two-phase fluid model where the flow field satisfies a non-Newtonian equation in the balk and the interface moves by the normal velocity given by the normal component of the flow field plus $\kappa$ times the mean curvature of the interface itself. The model can be formally obtained as the singular perturbation limit of a coupled system involving the Allen-Cahn equation and we are particularly interested in the existence and regularity issues of the weak solution in the setting of geometric measure theory. I describe what has been attempted in this direction with some recent advances, as well as the related problems.

Rayleigh-Taylor problem for incompressible two-phase flows with unmatched densities

HAO WU Fudan University Peoples Rep of China Co-Author(s):    Fei Jiang, Maoyin Lv, Hao Wu

Abstract: We investigate the Rayleigh-Taylor (RT) problem within the framework of diffuse-interface models for incompressible two-phase flows with unmatched densities under a uniform gravitational field. For the three-dimensional Abels-Garcke-Gr\{u}n system, we demonstrate that sufficiently large viscosity or mobility can inhibit the RT instability under appropriate stability conditions. Moreover, we establish a stability criterion that involves the system`s coefficients.