Free Boundary Problems 2026

7 Sept 2026, 08:55 → 11 Sept 2026, 16:00 Europe/Berlin

Humboldt Universität zu Berlin

Michael Hintermüller

17th International Conference on Free Boundary Problems: Theory and Applications 2026

The Free Boundary Problems (FBP) 2026 will take place in Berlin from September 7 to 11, 2026. The FBP conference is a flagship event that brings together the free boundary/partial differential equation community and is organized every few years with the most recent preceding conferences in the City of João Pessoa (Brazil, 2024), online in Berlin (Germany, 2021), Shanghai (China, 2017), Cambridge (UK, 2014) and Chiemsee (Germany, 2012) after the historical beginnings of the conference series in Montecatini (Italy, 1981).

In order to support young scientists, we provide a Young researcher Grant. PhD students can apply for a Young Researcher Grant, that waives the conference fee or provides accommodation. Applicants should send their CV and motivation letter to info@fbp2026.de no later than 30 June 2026.

 

FBP Legal Documents FBP 2026 AGBs Widerruf.pdf

Monday 7 September

10:00 → 11:15 Coffee Break

10:00 → 11:15 Registration

11:15 → 11:30 Plenary Talk: Welcoming and Opening

11:30 → 12:30 Plenary Talk: Martin Burger

Convener: Michael Hintermüller (Weierstrass Institute Berlin)

11:30 Phase Separation in Transport-Cross-Diffusion Models

This talk will give an overview of the apperance of phase-separation effects in nonlinear degenerate transpost-cross-diffusion models. Models of this kind can be found
in a variety of applications from physics over biology to social science. We will provide some examples and discuss the main mechanisms of competion between the degenerate diffusion and group-specific transport properties. Finally we discuss dense Brownian active particles and their analysis as a simple model system.

Speaker: Martin Burger

12:30 → 14:00 Lunch Break

14:00 → 16:00 Thin Material Structures: MS-11-1

Organisers: Weizhu Bao, Axel Voigt

Convener: Axel Voigt

14:00 Sharp Interface Models for Solid-State Dewetting Problems

In this talk, I will present sharp interface models with anisotropic surface energy for simulating solid-state dewetting and the morphological evolution of patterned islands on a substrate. We will show how to derive the sharp interface model via thermovariation dynamics, i.e. variation of the interfacial energy via an open curve with two triple points moving along a fixed substrate. The sharp interface model tracks the moving interface explicitly and it is very easy to be handled in two dimensions via arc-length parametrization. An efficient and accurate parametric finite element method (PFEM) was proposed for the sharp interface models. It is applied to study numerically different setups of solid-state dewetting including short and long island films, pinch-off, hole dynamics, semi-infinite film, tiny particle migration, etc. Our results agree with experimental results very well. In addition, extension to curved substrate and three dimensions will be discussed. Finally, we also present a reduced variational model via the Onsager's principle for small particle migration in solid-state dewetting. This is joint works with Wei Jiang, Yifei Li, David J. Srolovitz, Yan Wang and Quan Zhao.

Speaker: Prof. Weizhu Bao (National University of Singapore)

14:30 Convergence of finite elements for a bulk-surface coupled free boundary problem

In this talk, we present a numerical analysis of the Eyles-King-Styles tumor growth model, a free boundary problem coupling a Poisson equation in the bulk \Omega with a forced mean curvature flow on its boundary \Gamma. Unlike existing evolving surface analyses based on integer-order Sobolev spaces, this bulk-surface coupling requires H^{1/2}-order regularity on \Gamma. We establish a fractional Sobolev framework that admit a rigorous convergence analysis for continuous finite elements of polynomial degree at least three.

Speaker: Yifei Li (Tuebingen University)

15:00 A Multi-Phase-Field Approach to Solid-State Dewetting of Polycrystalline Thin Films

Solid-state dewetting is the process through which thin solid films break and retract on a substrate, leading to the formation of nanostructures. Dewetting in single-crystalline films is well understood as a surface-energy-driven phenomenon governed by surface diffusion. Polycrystalline films, by contrast, exhibit additional complexity due to the presence of extended defects (grain boundaries) forming between crystalline domains with different crystallographic orientations. To date, most theoretical and computational investigations have focused on the single-crystalline case. This presentation illustrates a grand-potential multi-phase-field model for simulating the dewetting of thin polycrystalline films. Assuming isotropic surface and interface energies, we demonstrate agreement with predictions based on energetic considerations and with the expected morphological evolution toward equilibrium. We further derive new analytical criteria for the onset of three-dimensional dewetting, providing fundamental theoretical benchmarks, and elucidate the key role of triple junctions in the dewetting dynamics. Finally, we investigate the dewetting behavior of finite polycrystalline patches, extending the scenarios previously established for single-crystalline films. Future perspectives towards including anisotropic surface and grain-boundary energies, as well as incorporating shear-coupled grain-boundary migration, are discussed.

Speaker: Marco Salvalaglio (TU Dresden)

15:30 Optimal L² error analysis of a loosely coupled finite element scheme for thin-structure interactions

Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluid-structure interaction (FSI) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions for the FSI problems in the standard L² norm. In this article, we propose a new stable fully-discrete kinematically coupled scheme for incompressible FSI thin-structure model and establish a new approach for the numerical analysis of FSI problems in terms of a newly introduced coupled non-stationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the L² norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field.

Speaker: Prof. Buyang Li (The Hong Kong Polytechnic University)

14:00 → 16:00 Free Boundary Problems in Data Science and Machine Learning: MS-13-1

Organisers: Leon Bungert, Eloi Martinet, Thorpe Matthew

Convener: Eloi Martinet (JMU Würzburg)

14:00 Adversarial training: non-local perimeter regularization

Recent work in machine learning has recognized that many standard algorithms for classification are strongly affected by adversarial attacks. Accordingly, a growing body of research has tried to identify ways to mitigate this issue. This talk will discuss a natural non-parametric formulation of this objective, which can be transformed into a standard classification problem that utilizes a non-local perimeter as a regularizer. I'll discuss recent work which i) establishes smoothness of classification boundaries under mild assumptions and ii) quantifies the degree to which adversarial attacks will modify the Bayes classifier. Connections with optimal transportation, mean curvature flow, and minimal surfaces, and related open problems will also be discussed.

Speaker: Ryan Murray (North Carolina State University)

14:30 Decomposability and extremality properties of nonlocal perimeters and variations

We focus on decomposability and extremality properties of nonlocal perimeters. Two archetypal types of these are the Gagliardo perimeter based on the eponymous seminorms and the nonlocal distributional Caccioppoli perimeter, both which can be considered with with finite and infinite interaction ranges.

A nonlocal notion of indecomposability associated to these perimeters is introduced, and we prove that it can be characterized solely in terms of the interaction range or horizon $\varepsilon$. Utilizing this, we show that it is possible to uniquely decompose a set into its $\varepsilon$-connected components, establishing a nonlocal analogue of the decomposition theorem of Ambrosio, Caselles, Masnou and Morel. This result is shown to apply also to Minkowski-type and adversarial perimeters.

Moreover, the extreme points of the balls induced by the Gagliardo and nonlocal total variation seminorm are identified, which naturally correspond to the two nonlocal perimeters. Surprisingly, while the extreme points in the former case are normalized indicator functions of $\varepsilon$-simple sets, akin to the classical TV-ball, in the latter case they are instead obtained from a nonlocal transformation applied to the extreme points of the TV-ball.

Speaker: José A. Iglesias (University of Twente)

14:00 → 16:00 Geometric Flows and Evolving Free Boundaries: MS-2-1

Organisers: Sebastian Hensel

Convener: Sebastian Hensel

14:00 The area preserving curve shortening flow with Neumann free boundary conditions

The area preserving curve shortening flow with Neumann free boundary conditions shortens a curve that is attached perpendicularly to a bounded domain $\Omega$ in the plane but keeps the enclosed area fixed. The end points of that curve move in the boundary curve of $\Omega$. Its limits are candidates for minimizers for the relative isoperimetric problem in the plane (thus, they are circular arcs). We describe the analytical nature of that flow and how the geometry helps to obtain results. Difficulties arise from the non-local nature of the flow. One recent result includes quantitative stability of critical points of the corresponding variational problem which is obtained with the help of the flow. Part of this talk is based on work obtained together with R. Neumayer, J. Park and M. Rupflin.

Speaker: Elena Mäder-Baumdicker (Freie Universität Berlin)

14:30 On the two-phase Mullins--Sekerka flow: Convergence and Existence

In this talk, we discuss up to two different aspects of the two-phase Mullins--Sekerka evolution.

In the first part, we investigate the convergence to equilibrium configurations: It is well-known that nearly spherical interfaces given as nearly radial graphs over spheres converge to an equilibrium configuration exponentially fast with a rate $\sim 1/R^3$, where $R$ is the radius of the corresponding equilibrium sphere. For increasingly large radii, this exponential rate deteriorates -- this reflects the fact that the spectral gap of the Laplacian vanishes as the period tends to infinity. We show how finer estimates of the dynamics of the flow along the center manifold of spheres allows for obtaining an algebraic convergence rate that persists even in the infinite-period-transition.

In the second part, which is based on ongoing joint work with Wenhui Shi, we might sketch how mild global-in-time solutions for the fully unbounded flow with initial conditions close to a hyperplane may be obtained.

Speaker: Saša Lukić (RWTH Aachen University)

15:00 De Giorgi varifold solutions to Volume Preserving Mean Curvature Flow

In this talk, I would like to introduce a novel weak solution concept for two-phase Volume Preserving Mean Curvature Flow, having both properties of unconditional global-in-time existence and weak-strong uniqueness. These solutions consist in evolving varifolds coupled with the phase volumes by a transport equation. We preliminarily concentrate on the existence, showing first that any sharp interface limit of solutions to a slightly modified nonlocal Allen-Cahn equation is a varifold solution according to this new definition. Then, we show, for the first time for a minimizing movements scheme, the unconditional convergence towards such a De Giorgi varifold solution, providing an alternative proxy for the completely degenerate $L^2$ distance. Finally, we introduce a new notion of volume-preserving gradient-flow calibrations to show that any classical solution to Volume Preserving Mean Curvature Flow is unique in the class of our new varifold solutions.

Speaker: Andrea Poiatti (University of Parma)

15:30 Quantitative homogenization of forced geometric motions through random fields of obstacles

Consider the evolution of sets by forced mean curvature flow through a field of random obstacles. The effective large scale behaviour is expected to be a first order motion. However, previous results heavily relied on the assumption that there is a global minimum speed of expansion and hence on the absence of any actual obstacles.

We obtain a quantitative homogenization result even with impenetrable obstacles, potentially allowing the interface to get stuck locally, eventually leading to enclosures behind a main front. So far in this regime not even a qualitative stochastic homogenization result had been available. The existence of a global minimum speed is replaced with a probabilistic assumption using the notions of "approximate stability" and an "effective minimum speed". This assumption is satisfied in particular for impenetrable obstacles distributed according to a Poisson point process with low enough intensity. The talk is based on joint work with Julian Fischer (ISTA).

Speaker: Jonas Ingmanns (Institute of Science and Technology Austria)

16:00 → 16:30 Coffee Break

16:30 → 17:30 Plenary Talk: Maria Giovanna Mora

Convener: Ulisse Stefanelli (Unviersity of Vienna)

16:30 On the Minimization of Nonlocal Interaction Energies

Nonlocal interaction energies play a central role in describing the collective behavior of large particle systems in a wide range of applications. In this lecture we will focus on interactions that are short-range repulsive and long-range attractive. We will review the key results on the existence and uniqueness of minimizers, and present their explicit characterization in the classical case of isotropic kernels with Riesz-type repulsion. We will then show how a complete characterization can be given for a broad class of anisotropic variants of this repulsive kernel. If time permits, we will conclude with a discussion of open problems and future directions.

Speaker: Prof. Maria Giovanna Mora (Università di Pavia)

17:30 → 18:45 Reception, Poster-Session, and Discussions

18:00 → 18:45 Plenary Talk: José Francisco Rodrigues

Convener: Michael Hintermüller (Weierstrass Institute Berlin)

18:00 Mathematics of Free Boundaries — A Brief History

The interdisciplinary field of free-boundary problems – that is, problems arising in a wide variety of phenomena in nature, science and technology where the boundaries delimiting the domains are not known a priori and constitute unknowns to be determined – saw remarkable developments in their mathematical treatment during the last half a century.

Seeking to contextualise this class of problems within a little-known historical perspective, which traces back to classical problems, such as the shape of a rotating drop or a body of minimum resistance in a fluid, the solidification of the Earth or the melting of ice and the formation of crystals, filtration in porous media to contact problems involving solids, the diffusion of oxygen in living tissues to pattern formation, image recognition to optimal timing in financial processes, the mathematical treatment of interfaces and free-boundary problems ranges from modelling to mathematical analysis and from numerical calculation to computational simulation.

This talks reviews some of these problems, with a focus on the obstacle problem, Stefan’s problem and other phase-change problems, highlighting in particular some examples of the most significant contributions presented at the triennial conference series “Free Boundary Problems: Theory and Applications”.

Speaker: José Francisco Rodrigues (Universidade de Lisboa)

Tuesday 8 September

09:00 → 10:00 Plenary Talk: Julian Fischer

Convener: Helmut Abels (Universität Regensburg)

09:00 Geometric relative energy methods for interface evolution problems and sharp-interface limits

The relative energy/entropy method has been a powerful tool for establishing (weak-strong) uniqueness, stability, and singular limits in continuum mechanics ever since its inception in the 70s by Dafermos and DiPerna. However, its application to interface evolution problems has faced a major obstacle in the form of the lack of strict convexity of the interface area functional. Recently, it has been shown that this difficulty can be overcome by developing an evolutionary analogue of the concept of calibrations for minimal surfaces. This has enabled the derivation of weak-strong uniqueness principles for curvature-driven evolutions without comparison principle, such as multiphase mean curvature flow or two-phase flow with surface tension, as well as corresponding quantitative convergence results for their diffuse-interface approximations.

Speaker: Prof. Julian Fischer

10:00 → 10:30 Coffee Break

10:30 → 12:30 Thin Material Structures: MS-11-2

Organisers: Weizhu Bao, Axel Voigt

Convener: Prof. Weizhu Bao (National University of Singapore)

10:30 Self-avoiding Fluid Deformable Surfaces

We propose a numerical method for fluid deformable surfaces governed by surface Stokes flow and Helfrich bending energy under active growth, aiming to model shape evolution of the epithelium sheets in developmental processes. As a new extension of the model, we prevent self-intersections, which commonly arise under large deformations or low enclosed volume to area ratios, by incorporating the nonlocal tangent-point energy to penalize non-embedded configurations. The resulting formulation is discretized using higher order surface finite elements similar to \cite{Krause2022}, with a parallelizable assembly strategy for the nonlocal terms. To tailor mesh quality to the geometric evolution, we propose a curvature-adaptive mesh redistribution strategy that improves mesh resolution in regions of high curvature. Numerical examples include the discocyte-to-stomatocyte transition, effectively traversing the well known minimizers of the Helfrich Energy, and the inversion of a sphere within a spherical confinement and thereby demonstrate the robustness of the method in capturing large deformations, self-avoidance, and growth-induced morphology changes.

Speaker: Maik Porrmann (Dresden University of Technology)

11:00 Numerical Simulation of Fluid Deformable Surfaces in Stream Function Formulation

We consider the surface Stokes-Helfrich problem using a stream function formulation. The formulation is considered for simply connected surfaces without boundary. It is based on a splitting of the velocity field in normal and tangential components and the Helmholtz decomposition of the tangential part. For its numerical solution the surface is approximated by higher order isoparametric elements and the resulting system is solved using the evolving surface finite element method (ESFEM). We systematically compare this approach with numerical solutions of the problem in velocity-pressure formulation using the established isoparametric Taylor-Hood element.

Speaker: Enno Igel (TU Dresden)

11:30 Vorticity-Homology and Variational Formulations for Navier--Stokes and Scriven Flows on Surfaces with Arbitrary Topology

Scalar vorticity formulation for fluid equations on surfaces is computationally attractive. However, the vorticity equation is incomplete on a non-simply-connected surface. We derive a new evolution equation for the finite dimensional harmonic (cohomology) components of the flow. We also show that the vorticity equation has a curvature-dependent, vorticity production term in addition to the advection-diffusion equation. The new terms finally make the vorticity formulation complete, and reveal structures that were previously overlooked. Specifically, they unify frictional boundary conditions, the Kutta condition, conservation law associated to isometry gauge, and a new conservation law the topological linking between the vortex geometry and streamlines of harmonic flows. In the limit of inviscid point vortex configuration, this new conserved quantity can be elegantly expressed in terms of the divisor class group when the vortices on surfaces are viewed as a divisor on a Riemann surface. These mathematical structures make these terms easy to incorporate computationally. They also lead to new nontrivial analytic solutions to the Euler equations.

In this talk, we also show that Scriven's equation for viscously evolving surface can be derived by Onsager's variational principle. This variational formulation survives after discretization, leading to a simple computational framework for Scriven's flow on arbitrary triangle mesh.

Speaker: Albert Chern (University of California San Diego)

12:00 Surface Beris-Edwards-Helfrich models - how local orientational order can influence global shape

We consider general models for hydrodynamic surface liquid crystals on (self-)evolving surfaces. We focus on nematic liquid crystals and model them using a Q-tensor approach. The model will be derived using the Lagrange-d´Alambert principle. Our Q-tensor is a 3D object defined on the surface. Here we address specific forms, essentially "surface conforming" Q-tensors, with eigenvectors in tangential and normal direction, and explore how special cases, like "flat degenerate" Q-tensors, with vanishing eigenvalue in normal direction, and the opposite case, with the dominating eigenvector pointing in normal direction and vanishing tangential Q-tensor, influence the bending properties of the surface. We demonstrate applications in biology for these special cases, as well as the general case and postulate a mechanical feedback mechanism based on these relations, which has the potential to drive shape evolutions.

Speaker: Axel Voigt

10:30 → 12:30 Free Boundary Problems in Data Science and Machine Learning: MS-13-2

Organisers: Leon Bungert, Eloi Martinet, Thorpe Matthew

Convener: Matthew Thorpe (University of Warwick)

10:30 Classification of large data with phase-field models

The analysis of Big Data is one of the most important challenges of the modern era. A first step in order to extract some information from a set of data is to partition it according to some notion of similarity. When only geometric features are used to define such a notion of similarity and no a priori knowledge of the data is available, we refer to it as the clustering problem.

Typically this labeling task is fulfilled via a minimization procedure. Of capital importance for evaluating a clustering method is whether it is consistent or not; namely it is desirable that the minimization procedure approaches some limit minimization method when the number of elements of the data set goes to infinity.

In this talk the consistency of a nonlocal anisotropic Ginzburg-Landau type functional for clustering is presented. In particular, it is proved that the discrete model converges, in the sense of Gamma-convergence, to a weighted anisotropic perimeter.

The talk is based on a work in collaboration with Matthew Thorpe (University of Warwick).

Speaker: Riccardo Cristoferi

11:00 Parametrizing Convex Sets Using A Sublinear Layer

We propose a single-layer neural parametrization of convex sets by learning sublinear (positively homogeneous and convex) functions. Our networks explicitly represent both the support and gauge functions of a convex body. We prove a universal approximation theorem for convex sets under this parametrization. Empirically, we demonstrate the method on shape optimization and inverse design tasks, achieving accurate reconstruction of target shapes.

Speaker: Eloi Martinet (JMU Würzburg)

11:30 Learning the Willmore flow with lightweight neural operators

The talk will focus on neural operators with few parameters for approximating the Willmore flow of oriented or non-oriented interfaces in space dimensions 2 and 3.
The proposed neural networks are trained on implicit representations of interfaces evolving by Willmore flow.
Various numerical simulations will be presented, together with applications to curve and surface reconstruction from point clouds.
This is joint work with Elie Bretin (INSA Lyon), Roland Denis (CNRS & U. Lyon 1), and Tokuhiro Eto (CNRS).

Speaker: Simon Masnou (Université Lyon 1)

12:00 Medial Axis Aware Learning of Signed Distance Functions

We present a variational neural approach for computing global signed distance functions (SDFs) from unoriented point clouds, focusing on the medial axis as the unknown jump set of the SDF gradient. The method is based on the observation that the SDF gradient is smooth away from the medial axis, but jumps where the nearest-point projection onto the surface is not well-defined. We formulate SDF reconstruction as a higher-order variational problem that enforces linear growth in the gradient direction away from the jump set, together with classical eikonal and zero-level set constraints. To model this (free) discontinuity set, we use an Ambrosio-Tortorelli type phase-field approximation, represented by a second neural network, to implicitly describe the medial axis and locally deactivate the higher-order regularization. Simultaneous optimization of the SDF and phase field yields accurate surface reconstruction, reliable far-field distances, and an interpretable phase-field representation of the medial axis, with numerical experiments showing improved performance over existing neural SDF methods.

Speaker: Samuel Weidemaier

10:30 → 12:30 Free Boundaries in Shape Optimization: MS-5-1

Organisers: Giuseppe Buttazzo, Matteo Novaga

Convener: Matteo Novaga (University of Pisa)

10:30 $2D$-lattice models for magnetic skyrmions under confinement

Starting from 2D lattice spin configurations satisfying the uniform state $\mathbf{e}_3$ outside a bounded domain $\Omega$, we investigate atomistic energies consisting of an exchange Heisenberg term together with an interfacial Dzyaloshinskii--Moriya interaction (DMI). In both discrete and continuum settings, imposing spatial confinement along with a topological constraint ($\text{degree}=1$) is essential to stabilize non-trivial localized skyrmions and prevent their collapse into the uniform state. To treat this constraint, we present two alternative strategies: introducing a non-degenerate discrete topological charge, or imposing an oscillation bound on the spins. The latter ensures that piecewise affine interpolations over a triangulation do not vanish, enabling their projection onto $\mathbb{S}^2$ and the standard prescription of the continuous degree. For sufficiently small DMI strength $\kappa$, both approaches lead to the existence of energy-barrier minimizers (lattice skyrmions) and their variational convergence, up to subsequences, to minimizers of the continuous energy (skyrmions in the continuum), rigorously linking atomistic chiral spin systems to continuum models in ultrathin ferromagnetic films.

Speaker: Maria Stella Gelli (Università di Pisa Dipartimento di Matematica)

11:00 A De Giorgi conjecture on the regularity of cartesian minimizers of the area in 1D

We discuss regularity properties of Cartesian minimizers of the (anisotropic) area functional
[
\int_a^b \Phi(-Du,1),dx+\int_a^b |u-g|^p,dx
]
defined on (BV(a,b)). We prove that if the (L^\infty)-norm of the forcing term (g) is sufficiently small, then every minimizer is locally Lipschitz continuous. Moreover, if the anisotropy (\Phi) is smooth and uniformly elliptic, then every minimizer is in fact of class (C^{1,1}). These results provide an anisotropic extension of a conjecture of De Giorgi concerning the regularity of Cartesian minimizers in dimension one and codimension one.

Speaker: Shokhrukh Kholmatov (University of Vienna)

11:30 Equilibrium shapes of liquid drops in the presence of discrete charges

In this talk I will present our treatment of a geometric variational
problem arising from modeling the equilibrium shapes of liquid drops
whose energy presents a competition of surface tension with the
repulsive Coulombic energy of a fixed number of point charges inside
the drop. The continuum analog of this problem in which the liquid is
treated as a perfect conductor is known to be variationally ill-posed,
hence the discrete nature of the charges preserved in our model
presents a non-trivial regularization whose properties are far from
obvious. In our model, we make a simplification of no dielectric
contrast between the liquid and its surroundings, which nevertheless
is an appropriate assumption for charged drops of liquid helium that
are used in applications to quantum chemistry. For large numbers of
charges, we identify a sharp charge threshold as the volume of the
drop goes to infinity jointly with the number of charges. This
threshold separates the regime of existence of minimizers from that of
non-existence and turns out to be considerably lower than the one
predicted by Rayleigh for continuum charge distributions, and below
the threshold the minimizer looks like a small perturbation of a ball
with charges distributed approximately uniformly over the drop
surface. Above the threshold, on the other hand, it is always
convenient to evaporate a single charge from the drop and move it to
infinity to lower energy.

Speaker: Cyrill Muratov (University of Pisa)

12:00 On the shape optimization for Hartree energies

We introduce a class of shape optimization problems modeled on Hartree type energies. From a shape optimization point of view, the energy is a sort of lower order perturbation of the first Dirichlet eigenvalue energy. We will partially discuss the existence and rigidity of optimizer in certain regimes and focus on the nonexistence issue in other regimes. The short talk is based on ongoing works with Dario Mazzoleni and Riccardo Moraschi (both from Pavia).

Speaker: berardo ruffini (Università di Bologna)

10:30 → 12:30 Phase Field Methods in Real-World Applications: MS-7-1

Organisers: Cecilia Cavaterra, Robert Lasarzik, Elisabetta Rocca, Hao Wu

Convener: Robert Lasarzik (WIAS)

10:30 Uniform higher order estimates for the sharp interface limit of a Navier-Stokes/Allen-Cahn system

We consider the linearized system for the sharp interface limit of a Navier-Stokes/Allen-Cahn system around a suitable approximate solution.
With the aid of a suitable weight taking the distance to the interface and the interfacial thickness into account we obtain optimal regularity estimates, which are uniformy in the interfacial thickness. This enables to improve previous convergence results in two space dimensions significantly and extend results to three space dimensions. This is a joint-work with Mingwen Fei, Yadong Liu, and Maximilian Moser.

Speaker: Prof. Helmut Abels (Universität Regensburg)

11:00 Homogenization of Viscoelastoplastic Flows in Randomly Perforated Domains

We study the homogenisation of an incompressible viscoelastoplastic fluid in randomly perforated domains. The mesoscopic model couples the balance of momentum with the evolution of a deviatoric internal stress tensor. The latter is transported by the flow through the Zaremba–Jaumann derivative and is subject to a non-smooth plastic dissipation law as well as stress diffusion. This model is used in geodynamics, for example, to describe the evolution of fault systems in the lithosphere under the influence of fluid flow over geological time scales.

The perforations are given by randomly distributed spheres with probable overlaps. Depending on the scaling of the radii against the microstructure scale, we examine different effective behaviour of the velocity and the internal stress.

This is joint work with Piotr Wozniak (FU Berlin) within project B09 "Materials with discontinuities on many scales" of CRC 1114 "Scaling Cascades in Complex Systems" funded by the German Research Foundation.

Speaker: Fan Cheng (Freie Universität Berlin)

11:30 Long-time dynamics of a convective Cahn--Hilliard model with dynamic boundary conditions

We consider a general class of convective bulk-surface Cahn-Hilliard systems with singular potentials. In contrast to classical Neumann boundary conditions, the dynamic boundary conditions of Cahn-Hilliard-type allow for dynamic changes of the contact angle between the diffuse interface and the boundary as well as absorption of material by the boundary. In this talk, I present recent results regarding the long-time behavior of weak solutions. In this context, we first show the existence of a minimal pullback attractor, and then, assuming a suitable decay on the velocity fields, we show that every weak solution converges as $t\rightarrow\infty$ to a single steady state.

Speaker: Jonas Stange (Universität Regensburg)

12:00 Global weak solutions to a diffuse-interface model for quasi-incompressible two-phase flows with unmatched densities and singular potential

In this talk, I will report a recent work on a thermodynamically consistent diffuse-interface model that describes the motion of two macroscopically immiscible, incompressible, and viscous Newtonian fluids with unmatched densities. This model is compatible with continuum mixture theory. It adopts a mass-averaged (barycentric) velocity so that the two-phase flow is quasi-incompressible: the velocity is no longer divergence-free, and the pressure enters the equation of the chemical potential. For the initial-boundary value problem in $\mathbb T^3$ with a class of physically relevant singular free energy densities, we prove the existence of global-in-time weak solutions. The proof relies on a suitable reduction of the original system to a Korteweg-type fluid model combined with a two-layer approximation, together with delicate estimates for the mass density and the phase-field variable inspired by the celebrated Bresch-Desjardins entropy. A key observation is that capillarity at the free interface provides a damping effect on the density evolution. For the limiting procedure, we derive delicate tail estimates to exclude possible concentrations of the singular potential, since no integrability of the pressure is available a priori. This work appears to be the first existence result for the Navier-Stokes/Cahn-Hilliard type system with unmatched densities and mass-averaged velocity without spatial regularization.

Speaker: Yadong Liu (Nanjing Normal University)

12:30 → 14:00 Lunch Break

14:00 → 16:00 Free Boundaries in Active Matter: MS-12-1

Organisers: Ricardo H. Nochetto, Shawn Walker

Convener: Shawn Walker (Louisiana State University)

14:00 Efficient Numerical Schemes for a Two-Phase Hydrodynamical Model of Active Liquid Crystals and Solids

We propose several linear, fully decoupled numerical schemes with first- and second-order temporal accuracy for a novel Q-tensor-based two-phase hydrodynamic model that describes the coupling of active nematic liquid crystal solutions with isotropic solid substrates. The model is derived from the generalized Onsager principle and includes nontrivial terms that contribute zero to the total free energy dissipation. We prove that the proposed decoupled linear schemes are thermodynamically consistent at the discrete level. In the passive limit, the SGE-BDF1 and SGE-PDG schemes are unconditionally energy stable, while the SGE-BDF2 scheme is energy stable with respect to a modified energy under a standard boundedness assumption and a sufficiently large stabilization parameter. We then perform extensive numerical simulations to investigate how activity and other model parameters affect active-nematic fluid- solid interactions. Finally, we analyze the physical mechanisms underlying the observed behaviors, providing deeper insight into the dynamics of soft, confined active nematic fluids.

Speaker: Qi Wang (U of South Carolina)

14:30 Instability of microbial droplets growing on viscous substrates

We develop and analyze a model for a flat microbial droplet growing on the surface of a viscous fluid. The model describes growth-induced stresses at the fluid surface, density variations in the bulk due to nutrient consumption, and the resulting fluid flows that arise. We reformulate this free boundary problem as a system of integro-differential equations defined solely on the microbial domain. From this formulation, we identify an axisymmetric solution corresponding to a radially expanding disk and analyze its stability. We find growth stabilizes the axisymmetric solution while buoyancy-driven flows destabilize it. Our analysis also leads to a spectral method for numerically solving the integro-differential equations on arbitrary smooth domains. We connect our findings to experimental observations of yeast growing on viscous substrates.

Speaker: Scott Weady (Flatiron Institute)

15:00 The Scalar Truesdell Time Derivative and (L²,H⁻¹)-Gradient Flows on Evolving Surfaces

Many mathematical models for active interfaces, biological membranes, and other free boundary problems combine geometric evolution with conserved quantities living on evolving surfaces. This motivates the development of general variational frameworks for deriving energy-dissipative evolution equations.

In this talk, we present a general variational framework for constructing coupled $(L^2,H^{-1})$-gradient flows on evolving surfaces. A key ingredient is the scalar Truesdell time derivative, which provides a natural evolution operator for conserved scalar quantities on moving manifolds. Combined with a compatible notion of surface variations, this yields evolution equations that simultaneously guarantee density conservation and energy dissipation.

Rather than focusing on a particular application, the framework serves as a general modeling principle for free boundary problems coupling geometric evolution with conserved surface fields. It systematically generates coupled evolution equations from a prescribed surface energy and naturally incorporates both normal and tangential surface motion. Several examples illustrate how classical geometric flows arise as special cases and demonstrate the versatility of the proposed framework for a broad class of coupled free boundary problems.

Speaker: Dr Ingo Nitschke (TU Dresden - Institute of Scientific Computing)

15:30 Stability and Bifurcations in Free Boundary PDE Models of Cell Motility

We begin with a brief overview of the rapidly developing research area of active matter, a.k.a. active materials. These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires the development of new mathematical tools. We present a free boundary PDE model a cytoskeleton of a moving cell. The key mathematical features of our model are the nonlocal boundary conditions, nonlinear diffusion, and the Keller-Segel cross-diffusion term. We present an overview of three recent works on that model. We begin from the 2D model with linear diffusion in which we derive an explicit formula for the stability determining eigenvalue for the linearized non-self-adjoint operator. Next, we present a recent result on the nonlinear stability of stationary and traveling wave solutions in 1D model. Here we focus on non-self-adjointness of the linearized problem, which plays a key role in the spectral stability analysis. Finally, we consider 2D model with nonlinear diffusion and prove this nonlinearity results in the change of the bifurcation from supercritical to subcritical, leading to two drastically different scenarios of the onset of the cell motion. Here we derive an explicit formula that governs the change of the bifurcation type in terms of measurable physical parameters and therefore can be used for both qualitative and quantitative biological predictions. Finally, we discuss how our results lead to an open question of bistability.

Speaker: Oleksii Krupchytskyi (The Pennsylvania State University)

14:00 → 16:00 Domain Walls and Patterns in Local and Non-local Geometric Variational Problems: MS-14-1

Organisers: Maria Stella Gelli, Cyrill Muratov

Conveners: Cyrill Muratov (University of Pisa), Maria Stella Gelli (Università di Pisa Dipartimento di Matematica)

14:00 Existence and Structure of 360 degree walls in Thin Ferromagnetic Films

We study the existence and structure of one-dimensional 360 degree walls in thin uniaxial ferromagnetic films. These structures are topological defects which consist of two oppositely charged 180 degree walls interacting by nonlocal stray-field energy. We show existence for nonzero wall angles when the stray-field interaction is sufficiently weak, while at zero angle no minimizer exists. We characterize scaling of the gruond state energy and the internal structure of the wall and obtain quantitative bounds on the separation of its two constituent sub-walls. This is joint work with A. Capella (UNAM) and C. Muratov (Pisa)

Speaker: Hans Knüpfer (University of Heidelberg)

14:30 Lipschitz regularity of almost-minimizers in one-phase problems with generalized Orlicz growth

Optimal local Lipschitz regularity for scalar almost minimizers of Alt-Caffarelli-type functionals
$$ \mathcal{F}({v}; \Omega) = \int_\Omega \varphi(x,\left|\nabla v(x) \right|)+ \lambda \chi_{\{{v}>0\}} (x) \,dx, $$ with growth function $\varphi$ a generalized Orlicz function, is established.
The results presented in this talk have been obtained in collaboration with Giovanni Scilla (Napoli), Francesco Solombrino (Lecce), and Anna Verde (Napoli).

Speaker: Chiara Leone (University of Naples Federico II)

15:30 Magnetic skyrmions beyond the conformal limit

In the radial setting, we analyze how magnetic skyrmions transition to (large scale) bubbles in the entire regime of positive energetic cost of domain walls, i.e., far away from the conformal limit. First, we demonstrate existence of radial, skyrmionic bubbles throughout this maximal regime of energetic feasibility. Second, we perform a Gamma-convergence analysis in the limit of vanishing domain wall energy and establish the precise leading order size and domain wall profile of the expanding bubbles.

This is joint work with Cyrill Muratov and Valeriy Slastikov.

Speaker: Theresa Simon (University of Münster)

14:00 → 16:00 Phase Transitions and Pattern Formation: MS-8-1

Organisers: Helmut Abels, Patrik Knopf

Conveners: Helmut Abels (Universität Regensburg), Patrik Knopf (Karlsruher Institut für Technologie (KIT))

14:00 Analysis of an Allen--Cahn system in two scale topology optimization

In this talk, we consider an Allen—Cahn system with the obstacle potential that guarantees mass conservation. This equation is coupled to two linear elasticity equations and a nonlocal operator. This system emerged from an algorithm for a problem in two-scale topology optimization using the phase-field approach. We prove the existence of weak solutions for the associated inclusion and comment on different connections of the solvability concept and the numerical algorithm. Numerical simulations show multi scale patterns in the resulting optimal structures.

Speaker: Robert Lasarzik (WIAS)

14:30 Stochastic phase-separation driven by transport noise

We propose a stochastic Cahn--Hilliard model driven by transport noise in order to describe phase-separation phenomena occurring in mixtures of
turbulent fluids. The model is analysed in its thermodynamically-relevant framework, namely employing a singular Flory--Huggins potential and a possibly degenerate mobility, and the noise is considered both in It\^o and Stratonovich form. As a first-step investigation, we establish well-posedness of the system in two and three spatial dimensions and its thermodynamical consistency. Several further developments are also discussed.

The works presented in the talk are based on joint collaborations with Andrea Di Primio (Scuola Normale Superiore, Pisa, Italy) and Andrea Papini (University of Gothenburg, Sweden).

Speaker: Luca Scarpa (Politecnico di Milano)

15:00 On the numerical treatment of the stochastic Cahn-Hilliard equation with singular potential

The Cahn-Hilliard equation is a deterministic model for the description of phase separation processes in metal alloys, which occur if the alloy is rapidly cooled below a critical temperature. This equation can be interpreted as an $H^{-1}$-gradient flow of the Ginzburg-Landau energy functional, which consists of a gradient term and double-well potential favoring phase separation. If the quench is shallow, i.e. the temperature is still close to the critical temperature, the double-well potential can be approximated by a smooth fourth-order polynomial. Yet, in the deep quench limit, i.e. when the temperature is significantly smaller than the critical temperature, a singular double-obstacle potential is the better choice. It is also well-known that in particular the early stages of the separation process are heavily influenced by thermal fluctuations which are not included in the deterministic description.
In this talk, we discuss the numerical treatment of the stochastic Cahn-Hilliard equation with double-obstacle potential and conservative noise on a periodic domain. In particular, we propose a fully discrete finite element scheme and present a convergence result. In this endeavor, special attention has to be paid to the interplay between the singularities of the potential and the stochastic forcing term. In comparison to the stochastic Allen-Cahn equation, which is based on an $L^2$-gradient flow, balancing these challenges in the case of the Cahn-Hilliard equation poses additional difficulties due to the underlying $H^{-1}$-structure. Conceptually, our proof relies on monotonicity arguments and omits the application of Skorokhod's theorem, which allows us to show convergence towards probabilistically strong solutions.
We conclude by presenting numerical simulations underlining the practicality of the proposed scheme and the importance of the additional stochastic fluxes.

Speaker: Stefan Metzger (Friedrich-Alexander-Universität Erlangen-Nürnberg)

15:30 Anisotropic crystal growth on surfaces

We consider a phase field model for crystal growth on a curved surface.
A particular emphasis must be placed on the choice of the anisotropic
energy density functional. We advocate for a construction that is based on
fixing a density on the tangent space of a chosen point on the surface,
and then moving it along geodesics to the other tangent spaces.
We propose a surface finite element method that is unconditionally stable
and present some numerical results, including for the modelling of ice
crystal growth on a sphere.

Speaker: Robert Nürnberg

16:00 → 16:30 Coffee Break

16:30 → 18:30 Contributed Talks: CS-1

Convener: Michael Eden (University of Regensburg)

16:30 A Stokes-Reaction System with Evolving Microstructure

We consider a coupled Stokes–reaction–diffusion system posed in a non-periodically perforated domain with solid spherical inclusions whose radii evolve according to a surface reaction law. In the critical scaling regime, where the inclusions have size of order (\e^3), we prove well-posedness of the microscopic problem using a contraction mapping argument. We then analyze the asymptotic asymptotic limit as (\e\to0). The resulting effective model couples a Brinkman-type flow equation to a macroscopic reaction–diffusion equation and a transport equation for the measure-valued particle-size distribution.

Speaker: Michael Eden (University of Regensburg)

eden_fbp26.pdf

17:00 Nonlinear stability of self-similar shrinkers in mean curvature flow

Self-similar shrinkers describe generic singularities of multiphase mean curvature flow at the parabolic scale. The classification of stable self-shrinkers, conjectured by Ilmanen, is a central step towards understanding global dynamics of generic flows. While the compact case of the circle is well understood at this point, classification of stable, non-compact shrinkers remains still open.

In this talk, we present a proof of the quantitave, nonlinear, local stability of the lens and three-ray star in terms of Huisken‘s F-entropy, completing the classification by covering all possible cases. The argument is based on a calibration technique together with a rigorous linearisation of the F-entropy. This is joint work with Julian Fischer and Theresa Simon.

Speaker: Lauro Silini

16:30 → 18:30 Numerical Methods for Geometric PDEs: MS-3-1

Organisers: Buyang Li, Robert Nürnberg

Convener: Buyang Li (The Hong Kong Polytechnic University)

16:30 Boundary Shape Derivatives with Unfitted Finite Elements

We present a framework for computing shape derivatives of boundary functionals discretized with unfitted finite element methods. The main idea is to replace the boundary functional by a narrow-band volumetric regularization prior to discretization. This yields an exact Fr'echet derivative of the resulting discrete functional with respect to the discrete level set function, without any mesh-related restrictions.

At the continuous level, we establish Fr'echet differentiability of the regularized boundary functional and show that, as the regularization thickness converges to zero, both the functional and its derivative converge to the classical boundary shape functional and shape derivative.
At the discrete level, we prove optimal-order error estimates with respect to the mesh size for the functional and for its Fr'echet derivative that are independent of the regularization thickness. As a result, the regularized formulation admits consistent and stable discrete gradients suitable for gradient-based shape optimization on unfitted meshes with fixed regularization.

The approach accommodates higher-order level set representations and extends, via standard Lagrangian techniques, to PDE-constrained problems.
Numerical experiments confirm the exactness of the discrete derivative, the predicted convergence rates, and the applicability of the method to geometric optimization problems.

Speaker: Shawn Walker (Louisiana State University)

17:00 Curvature tensors for piecewise smooth metrics

If a simplicial triangulation is equipped with a piecewise smooth Riemannian metric that has single-valued tangential-tangential components on element interfaces, then there are various notions of curvature that one can define, even though the classical formulas for curvature involving derivatives of the metric no longer make sense. In this talk, I will explain the origins of these definitions and summarize what is known about the convergence of these discrete curvatures under refinement of the triangulation. The curvatures that I will discuss include the scalar curvature, Einstein curvature tensor, and Riemann curvature tensor. This is joint work with Yakov Berchenko-Kogan and Michael Neunteufel.

Speaker: Evan Gawlik (Santa Clara University)

17:30 Blow-up finite elements for tensor fields on surfaces

Blow-up finite elements, developed jointly with Evan Gawlik, were motivated by a vexing problem when discretizing tangent vector fields on surfaces: For a discretized surface, the angles at vertices generally no longer sum to 360 degrees. As a result, it is not possible to construct a vector field approximation that is continuous within each element, tangent to the surface, and continuous across each edge (in the sense that there is no jump in the component tangent to the edge and no jump in the component normal to the edge). Previous approaches either broke tangentiality to the surface or continuity across edges. With blow-up finite elements, we can keep both of these properties by allowing the vector fields to vary rapidly near vertices. The resulting vector fields are both tangent to the surface and single-valued on edges, but they are multi-valued at vertices. I will define these elements for vector fields and tensor fields, discuss some preliminary numerical results, and discuss potential applications to numerical geometry and to intrinsic discretization of the surface Stokes equations for creeping flow.

Speaker: Yakov Berchenko-Kogan (Florida Institute of Technology)

16:30 → 18:30 Free Boundaries in Shape Optimization: MS-5-2

Organisers: Giuseppe Buttazzo, Matteo Novaga

Convener: Giuseppe Buttazzo (University of Pisa)

16:30 Spectrum of the p−Laplacian on general open sets.

We present a variational definition of the essential spectrum suitable for nonlinear operators such as the Dirichlet p-Laplacian and discuss an extension of Persson's theorem to this framework. Finally, we show that the Ljusternik-Schnirelmann minmax levels of the constrained p-Dirichlet integral are attained whenever one of these levels is below Persson's threshold. This is based on joint works with Lorenzo Brasco, Giovanni Franzina, and Francesca Prinari.

Speaker: Luca Briani (Technische Universität München)

17:00 Fourth order eigenvalue problems in geometry

Motivated by the study of the Paneitz operator in geometry, we consider eigenvalue problems for some fourth-order problems, with a particular emphasis on eigenvalue bounds and shape optimization.

Speaker: Maria del Mar Gonzalez Nogueras (Universidad Autonoma de Madrid)

17:30 Boundary regularity of a fourth-order free boundary problems.

Consider the following functional
$$u\in H^2(D,\mathbb{R})\mapsto\int_D(|\nabla^2 u|^2+\chi_{u\neq 0})$$ where $D\subset \mathbb{R}^2$. This is a higher order analogue of the Alt-Caffarelli problem, and its local minimizers are linked to several shape optimization question: primarily with the minimization (under area constraint) of the critical buckling load of a clamped plate $\Omega\subset\mathbb{R}^2$, defined as $$\Lambda(\Omega):=\inf_{u\in H^2_0(\Omega,\mathbb{R})\setminus\{0\}}\frac{\int_{\Omega}|\nabla^2 u|^2}{\int_{\Omega}|\nabla u|^2},$$ as well as the minimization of the drag of an obstacle with fixed measure in a Stokes fluid. I will give a description of the free boundary, which is expected to be a union of regular curves joined with an angle of $102.5°$.

This is a joint work with Jimmy Lamboley.

Speaker: Mickaël Nahon (Université Grenoble Alpes)

18:00 From diffuse dislocations to polygonization: a Beltrami-driven instability

We propose a mechanism for the onset of polygonization in crystals with dislocations. Starting from a diffuse distribution, we show that the interaction between elasticity and dislocation motion can destabilize the homogeneous state. The classical Peach–Koehler force tends to stabilize this state, while a non-local Beltrami contribution drives the instability.
The fastest-growing structures are oriented at 45^\circ, although no characteristic spacing is selected at this stage. Two numerical simulations illustrate the formation of these patterns and their possible evolution toward polygonized dislocation walls under bending.

Speaker: Nicolas Van Goethem (Universidade de Lisboa)

16:30 → 19:00 Phase Field Methods in Real-World Applications: MS-7-2

Organisers: Cecilia Cavaterra, Robert Lasarzik, Elisabetta Rocca, Hao Wu

Convener: Robert Lasarzik (WIAS)

16:30 Optimal control of a tumor growth model with hyperbolic relaxation of the chemical potential

In this talk, we study the optimal control of a phase field system of viscous Cahn-Hilliard type modeling tumor growth, in which a hyperbolic relaxation of the chemical potential is added to the equation governing the phase evolution. In addition to well-posedness results for the state system, we analyze the differentiability properties of the associated control-to-state operator and derive first-order necessary optimality conditions for the control problem. If time permits, we also provide some asymptotic results as the hyperbolic relaxation term tends to zero. This is a joint work with P. Colli and E. Rocca from Pavia.

Speaker: Jürgen Sprekels (Weierstrass Institute, Anton-Wilhelm-Amo-Strasse 39, 10117 Berlin, Germany)

17:00 The Cahn–Hilliard Equation with Nonlinear Diffusion Beyond Convexity

In this talk, we consider a class of Cahn–Hilliard equations with nonlinear diffusion, arising in the description of complex materials.
We discuss recent results on existence, regularization, stability, and convergence to equilibrium under very general assumptions. In particular, we show that the strong convexity assumption on the interfacial energy, which underlies much of the existing theory, can be removed while still obtaining a comprehensive description of the dynamics.

Speaker: Monica Conti

17:30 Analysis and coarsening dynamics of the Active Cahn-Hilliard equation

In this talk I will present new results about the Active Cahn-Hilliard equation, which is a variant of the Cahn-Hilliard equation not derivable from variational principles which describes active materials constituted by particles that can convert energy into directed motion. This model has multiple applications in biomedicine and engineering. I will show quantitative results about the phase-separation and coarsening dynamics, and both analytical and numerical results for the model equation.

Speaker: Abramo Agosti (University of Pavia)

18:00 Split-step algorithms for generalized gradient systems in phase-field models

Several phase-field models have the underlying structure of
generalized gradient systems in Banach spaces, whose evolutions
are generated by the interplay between an energy functional
and a dissipation potential. We focus on the case in which the dual
dissipation potential is given by a sum of two functionals and show
that solutions of the associated gradient-flow evolution equation with
combined dissipation can be constructed by a split-step method, i.e. by
solving alternately the gradient systems featuring only one of the
dissipation potentials and concatenating the corresponding
trajectories. Thereby the construction of solutions is provided either by
semiflows, on the time-continuous level, or by using Alternating Minimizing
Movements in the time-discrete setting. In both cases the convergence
analysis relies on the energy-dissipation principle for gradient systems.

Joint work with Alexander Mielke (Berlin) and Artur Stephan (Vienna).

Speaker: RICCARDA ROSSI (Università degli studi di Brescia)

18:30 A multiphase model for saline droplets

We introduce an isothermal phase-field model for the coupled evaporation and crystallization of an aerosol droplet containing dissolved solutes: as the liquid evaporates, the dissolved solute, for example salt, becomes increasingly concentrated and, once it exceeds a saturation threshold, precipitates by forming a crystalline phase.
The model is described by a three-phase (liquid, crystalline, vapor) Cahn-Hilliard/Allen-Cahn system. The evolution of the phases is coupled with a diffusion equation for the solute concentration.
A central modeling choice is to consider the dissolved or crystalline solute content as a conserved order parameter while allowing the liquid to become supersaturated with solute. The saturation concentration is imposed softly, namely, supersaturation is permitted, but carries an energetic cost that increases smoothly beyond the saturation threshold.
We derive and study such a model based on a thermodynamic structure.

Speaker: Erica Ipocoana (Freie Universität Berlin)

Wednesday 9 September

09:00 → 10:00 Plenary Talk: Joaquim Serra

Convener: Damião J. Araujo (Universidade Federal da Paraíba)

09:00 Regularity of Stable Free Boundaries

In many nonconvex variational free-boundary problems, regularity theory has traditionally focused on energy minimizers. Yet stable—and, more generally, finite-index—critical points arise naturally in physical models and geometric variational constructions, and are often just as relevant. A basic question is therefore: how much of the regularity theory for minimizing free boundaries survives beyond minimizers?

Such an extension is far from automatic: many arguments for minimizers rely on comparison principles or energy-decreasing deformations that are unavailable for general critical points. I will explain the significance and main difficulties of this question, and present recent progress for the free-boundary analogue of the Allen–Cahn equation in dimension three and for the one-phase Bernoulli problem in dimensions three and four.

This talk is based on joint work with X. Fernández-Real, partly in collaboration with H. Chan and A. Figalli.

Speaker: JOAQUIM SERRA (ETH ZURICH)

10:00 → 10:30 Coffee Break

10:30 → 12:30 Contributed Talks: CS-2

Convener: Malte Kampschulte (Charles University Prague)

10:30 From fluid structure interaction towards Navier-Stokes-Vlasov

Consider the coupled system of rigid particles flowing inside a Navier-Stokes fluid. If one sends the number of particles to infinity while at the same time shrinking their size, then at the right scaling the expected limit is the Navier-Stokes-Vlasov system. Turning this formal limit into a rigorous proof however is still an open problem. The aim of this talk is to present some partial results under additional assumptions of controls on particle distance. The key to this is understanding the underlying fluid-structure interaction problem in a free boundary sense and then dealing with the boundary stresses as a direct part of the equation instead of a merely another coupling condition of the interaction. This is based on joint work with Richard Höfer (Regensburg).

Speaker: Malte Kampschulte (Charles University Prague)

11:00 Finite element analysis of a coupled Navier–Stokes flow -- linear elasticity problem

This talk presents the study on the finite element approximation of a fluid–structure interaction (FSI) problem. The FSI problem is governed by the incompressible Navier–Stokes equations and the equations of linear elasticity.
We consider that both the fluid and solid subdomains are stationary. We incorporate the grad-div stabilization for the fluid part during the spatial discretization. In the time continuous case, we show the existence of a finite element solution, and derive the a priori estimates and strong a priori estimates. We will present the finite element error analysis and derive the error estimates. The derived error estimates are independent of Reynolds number.

Speaker: Mr Nishant Ranwan (PhD scholar at School of Mathematics, IISER Thiruvananthapuram)

11:30 Well-posedness and Conditional Regularity for Compressible Fluid--Shell Interaction

The interaction of a three-dimensional barotropic compressible fluid with a viscoelastic shell occupying part of the fluid boundary is considered. Local-in-time existence and uniqueness of strong solutions are established in a fully Eulerian framework based on a localised Hanzawa transform. In addition, blow-up criteria are derived, providing sufficient conditions for extending strong solutions beyond a potential singular time.

Speaker: Pierre Marie Ngougoue (University of Duisburg-Essen)

12:00 No-contact results for fluid-plate interaction

Motivated by the problem of rigorously justifying reduced elastohydrodynamic models, which are commonly used in microfluidics, we study a fluid–plate interaction system and identify conditions guaranteeing uniform separation of the elastic plate from the opposing rigid boundary.

More precisely, we consider a three-dimensional incompressible viscous fluid governed by the Navier–Stokes equations beneath an elastic plate governed by a fourth-order equation, possibly with square-root structural damping. The two subsystems are coupled through velocity- and stress-matching conditions. We present two sufficient conditions that exclude contact between the plate and the rigid boundary. First, sufficiently small initial kinetic and elastic energies relative to the square of the conserved mean fluid height guarantee a uniform positive lower bound on the plate height and, consequently, global-in-time existence of weak solutions [1]. Second, natural energy bounds together with suitable additional regularity of weak solutions yield uniform control of the reciprocal plate height, without any smallness assumption and even in the absence of structural damping [2].

These results remove a key obstacle to applying rigorous lubrication-approximation techniques used to justify sixth-order thin-film equations as reduced elastohydrodynamic models.

This is joint work with Igor Kukavica (University of Southern California), Linfeng Li (University of California Los Angeles), and Boris Muha (University of Zagreb).

References:

[1] M. Bukal, I. Kukavica, L. Li, and B. Muha. A global existence result on weak solutions for the 3D Navier–Stokes–plate system with no contact. J. Nonlinear Sci. 36, 60 (2026).

[2] M. Bukal, I. Kukavica, L. Li, and B. Muha. A no-contact result for a plate–fluid interaction system in dimension three. To appear in SIAM J. Math. Anal. (2026).

Speaker: Mario Bukal (University of Zagreb Faculty of Electrical Engineering and Computing)

Bukal_FBP26.pdf

10:30 → 12:30 Numerical Methods for Geometric PDEs: MS-3-2

Organisers: Buyang Li, Robert Nürnberg

Convener: Robert Nürnberg

10:30 Minimal horizontal triods

In this talk we discuss the question of finding a network configuration of minimal length connecting three given points in the Heisenberg group.

After formulating a suitable horizontal curve shortening flow, we present numerical experiments based on a stable fully discrete finite element scheme that provide useful insights into the rich landscape of this sub-Riemannian geometry.

This is joint work with Robert Nuernberg.

Speaker: Paola Pozzi

11:30 Error analysis of a nonconforming surface finite element method for the vector Laplacian

Nonconforming surface finite elements have recently been developed to discretize 3D vector-valued compressible flow problems arising in climate modeling. In this talk, we present an error analysis of this approach for a vector-valued Laplace problem, a key operator in fluid equations on surfaces. The problem is discretized via edge-integration on local flat triangles using the nonconforming linear Crouzeix–Raviart element, which is continuous at edge midpoints in each vector component. We first introduce the vector-valued Laplace problem on the surface and its Crouzeix–Raviart discretization. We then present interpolation estimates and derive optimal error bounds in the H^1- and L^2-norms. Finally, we show numerical experiments validating the theoretical convergence rates.

Speaker: Carolin Mehlmann

12:00 Viscoelastic two-phase flows

The numerical simulation of viscoelastic two-phase flows involves complex free boundary dynamics and faces major challenges, such as the High Weissenberg Number Problem and the loss of positive definiteness of the conformation tensor.
In this presentation, we introduce an energy-stable numerical framework designed to address these issues. First, we discuss energy-stable, positivity-preserving discretizations for viscoelastic fluid models and highlight novel convergence results. Second, we extend these approaches to the two-phase setting using a parametric finite element method (PFEM) for the coupled bulk-interface system, ensuring unconditional solvability and energy stability. We discuss the preservation of physical and geometric structures by incorporating global Lagrange multipliers to guarantee exact volume conservation and energy dissipation. We demonstrate the robustness of the proposed methods with numerical experiments.

Speaker: Dennis Trautwein (Universität Regensburg)

10:30 → 12:30 Hydrodynamic Models with Moving Contact Lines: MS-6-1

Organisers: Hans Knüpfer, Lorenzo Giacomelli, Dirk Peschka

Conveners: Dr Dirk Peschka (Weierstrass-Institut für Angewandte Analysis und Stochastik), Hans Knüpfer (University of Heidelberg), Lorenzo Giacomelli (Sapienza University of Rome)

10:30 Sharp-Interface Modeling of Dynamic Wetting with a Contact-Region Generalized Navier Boundary Condition

Dynamic wetting poses a fundamental modeling challenge because the motion of a material contact line requires slip at the solid wall and a consistent coupling between contact-line kinematics, wall stresses, and the dynamic contact angle. After briefly reviewing these modeling constraints, we present the contact-region generalized Navier boundary condition (CR-GNBC). The model replaces the singular uncompensated Young stress at the contact line by a smooth distribution over a finite contact region, whose width is treated as a physical parameter independent of the computational mesh. The resulting sharp-interface formulation is consistent with the kinematic evolution of the contact angle and avoids prescribing the dynamic angle through an empirical contact-line velocity law.

On the numerical side, we discuss the main ingredients required for direct simulations of wetting processes within sharp-interface finite-volume approaches. Particular attention is given to the accurate representation and transport of the interface and contact line, the numerical treatment of wall and contact-line conditions, and the verification of mesh-independent solutions. Representative simulations illustrate the regularizing effect of the CR-GNBC and its potential for predictive numerical modeling of dynamic wetting.

Speaker: Dieter Bothe

11:00 Dynamics of a 3d rimming-flow equation

We study the dynamic behaviour of a thin viscous fluid film coating the inner wall of a rotating cylinder -- a so-called rimming flow.
The resulting equation for the height of the fluid film is a quasilinear, degenerate parabolic PDE of fourth order.
Various competing effects drive the dynamics of the interface -- viscosity, surface tension and gravity.
For a positive surface tension parameter of order one, we investigate existence and stability of positive steady states, depending on the influence of gravity.

Speaker: Christina Lienstromberg (University of Stuttgart)

12:00 Asymptotic behavior of solutions to the nonlinear degenerate PDEs describing evolution of thin viscous sheets

In this talk, we consider the nonlinear system of coupled degenerate PDEs describing evolution of the free surface of a viscous thin liquid sheet. We show that in the associated Lagrangian coordinates the system transforms to the singular diffusion porous medium type equation equipped with the non-homogeneous in space and time source term having zero mean.
In the regime of vanishing sheet surface tension, we show H^1-exponential asymptotic decay to the flat profile of the solutions to the latter equation considered with general initial data in spatial dimensions d<=3. The results were obtained in joint work with Marco Fontelos and Roman Taranets.

Speaker: Georgy Kitavtsev (Middle East Technical University, Nothern Cyprus Campus)

10:30 → 12:30 Phase Transitions and Pattern Formation: MS-8-2

Organisers: Helmut Abels, Patrik Knopf

Conveners: Helmut Abels (Universität Regensburg), Patrik Knopf (Karlsruher Institut für Technologie (KIT))

10:30 Phase-field models for growing viscoelastic solids

Accretive growth is a fundamental mechanism in biological, natural, and technological systems. I will present two phase-field models for finite-strain viscoelastic solids undergoing mechanically driven accretion, addressing both phase-transition dynamics and the build-up of growth-induced incompatibilities.

The first model describes a two-phase viscoelastic medium in which one phase irreversibly accretes at the expense of another, with growth governed by a generalized eikonal equation coupled to quasistatic viscoelastic equilibrium. Both diffuse- and sharp-interface formulations are considered, and existence of weak/viscosity solutions is established together with energy balances and sharp-interface convergence.

The second model includes the description of incompatible growth through a multiplicative decomposition of deformation into elastic strain and backstrain, modeling the residual stresses generated as newly accreted material is deposited in an unstressed state. A compliant fictitious surrounding medium regularizes the free-boundary problem, enabling the proof of existence for the fully coupled system.

This is joint work with Andrea Chiesa (U. Vienna).

Speaker: Ulisse Stefanelli (Unviersity of Vienna)

11:00 Discretization of Cahn--Hilliard equations with dynamic boundary conditions: error estimates and adaptivity

In this talk we analyse a bulk--surface finite element in space and backward difference in time full discretization of a general two-parameter family of Cahn--Hilliard equations with dynamic boundary conditions. A novel proof strategy using discrete almost mass conservation and a suitable Poincaré–Wirtinger inequality is presented to achieve optimal-order fully discrete error estimates in the $L^2$- and $H^1$-norm. Additionally, we present an adaptive algorithm in time and space with estimators originating from a residual based a posteriori error analysis. Numerical examples illustrate and validate the theoretical results.

Speaker: Nils Bullerjahn

11:30 Gradient flows of degenerate viscoelastic phase separation

Viscoelastic phase separation occurs in binary fluids where the constituent molecules aggregate on strongly different time scales. Typically, the morphology of this process features volume shrinking, sponge-like structures and phase inversion. Such phenomena are observed in polymer solutions, for instance, where polymer chains are much larger and migrate much more slowly compared to solvent molecules.

In this talk, we consider a dissipative diffuse-interface model, introduced by Zhou, Zhang and E (2006), which couples a Cahn–Hilliard equation for the phase variable with the spherical part of the bulk stress, governed by relaxation dynamics. Our main objective is to establish existence of weak solutions and (conditional) weak–strong uniqueness, assuming degenerate mobility functions and possibly singular potentials. The analysis follows a gradient flow approach with an underlying metric of Benamou–Brenier (Wasserstein) type that reflects a coupled system of two equations. Two aspects prove particularly challenging: the absence of a classical Euler–Lagrange equation and the task of establishing convexity of the internal energy functional.

Speaker: Moritz Gau (WIAS Berlin)

12:00 Local Well-posedness of a Diffuse Interface Two-phase Flow Model with Surfactants

We establish the local well-posedness of strong solutions for a thermodynamically consistent diffuse interface model describing two-phase flows with surfactants, specifically the so-called Model C introduced by Garcke, Lam, and Stinner. The highly non-standard, mixed-order strongly coupled structure of Model C invalidates the classical Agmon-Douglis-Nirenberg theory and conventional Lopatinskii-Shapiro boundary conditions due to divergent scaling behaviors. To overcome this, we develop a tailored resolvent analysis by introducing an augmented source term. Furthermore, we introduce and analyze a structural variant, denoted as Model D, which achieves a crucial structural decoupling by transferring the interfacial surfactant energy entirely from the gradient part to the potential part, and we readily establish its maximal $L^p$-regularity.

Speaker: Songzhuang Chen

12:30 → 14:00 Lunch Break

14:00 → 16:00 Contributed Talks: CS-3

Convener: Konstantinos Bessas (University of Pavia)

14:00 Generalized BMO-type seminorms and vector-valued Sobolev functions

We establish a pointwise limit theorem for a broad class of parameter-dependent BMO-type seminorms as the parameter tends to zero. By introducing novel BMO-type seminorms, we provide a unified framework that extends several existing results and yields derivative-free characterizations of Sobolev-type spaces, both in the scalar and in the vector-valued setting.
More precisely, for any open set $\Omega\subset \mathbb{R}^n$ and any $p\in (1, \infty)$, we provide a characterization of the Sobolev space $W^{1,p}(\Omega; \mathbb{R}^m)$. In addition, we characterize the space $E^{1,p}(\Omega;\mathbb{R}^n)$ of $L^p$ maps with $p$-integrable distributional symmetric gradient.
Finally, for all $p\in [1, \infty)$, we show that these seminorms converge to integral functionals with convex, $p$-homogeneous integrands associated with the distributional gradient and the symmetric gradient.

Speaker: Dr Konstantinos Bessas (University of Pavia)

14:30 On a stochastic phase-field model of cell motility with singular diffusion

We study existence of solutions in the variational sense for a class of stochastic phase-field models describing moving boundary problems. The models consist of stochastic reaction-diffusion equations with singular diffusion forced by a phase-field. We investigate both the case of an independently evolving phase-field and of coupled phase-field evolution driven by a viscous Hamilton-Jacobi equation. Such systems are used in the modelling of single-cell chemotaxis, where the contour of the cell shape corresponds to a level set of the phase-field. The technical challenge lies in the singularities at zero level sets of the phase-field. For large classes of initial data, we establish global existence of probabilistically weak solutions in $L^2$-spaces with weights which compensate for the singularities.

Speaker: Amjad Saef

15:00 Analysis of a Surface Crouzeix-Raviart Element for the Stokes Problem

Surface flow problems have attracted considerable attention in recent years because of their broad applications in science and engineering. In this talk, we investigate the numerical approximation of the surface Stokes equations posed on a two-dimensional manifold embedded in three-dimensional space. The surface velocity is discretized using the nonconforming Crouzeix–Raviart finite element, while the pressure is approximated by piecewise constant functions on a polyhedral approximation of the surface. A fundamental challenge stems from the discretization of the symmetric surface strain-rate tensor, whose kernel contains non-physical velocity modes when combined with the Crouzeix–Raviart element. To address this issue, we introduce a stabilized formulation of the momentum equation and establish the discrete inf-sup stability of the resulting finite element pair with respect to a stabilization-dependent energy norm. We further derive optimal a priori error estimates in both the energy and $L^2$ norms. The analysis is particularly challenging due to the interplay between the symmetric strain-rate tensor, the nonconforming velocity approximation, the pressure variable, and geometric consistency errors arising from the discrete surface. We discuss the key analytical ingredients used to overcome these difficulties and prove the convergence results. Finally, numerical experiments confirm the predicted convergence rates and demonstrate the accuracy and robustness of the proposed method.

Speaker: Manisha Chowdhury (Otto-von-Guericke-University Magdeburg)

15:30 The generic structure of the singular set in the Stefan problem

The Stefan problem describes the evolution of phase transitions, such as ice melting into water. In this talk, we will discuss the fine structure of the singular set of the free boundary. More precisely, we recently proved that, outside an $(n-2)$-dimensional set, the singular part is contained in an $(n-1)$-dimensional manifold of class $C^\infty$. We will first describe this result and explain its implications for the generic-in-time regularity of free boundaries in low dimensions. Then, we will present recent joint work with Alessio Figalli showing that, for a generic solution, the whole singular set has parabolic dimension at most $n-2$.

Speaker: Giacomo Colombo (ETH Zürich)

14:00 → 16:00 Free Boundaries in Active Matter: MS-12-2

Organisers: Ricardo H. Nochetto, Shawn Walker

Convener: Shawn Walker (Louisiana State University)

14:00 Patterns in active nematic films

I will discuss pattern formation in active matter confined to a thin film of microtubules activated by Adenosine Triphosphate (ATP), focusing on configurations corresponding to high anisotropy of the elastic constants. This is the result of the joint work with M. Carme Calderer (Minnesota), Jordi Ignes-Mullol (Barcelona), Francesc Sagues (Barcelona), Lingxing Yao (Akron), and Longhua Zhao (Case Western).

Speaker: Dmitry Golovaty

14:30 Defects, geometric frustration and free boundaries in liquid crystals

Liquid crystals provide remarkable examples of two-way coupling between internal order and geometry: orientational or layered order responds to confinement and interfaces, while the resulting elastic stresses can reshape the domain itself. I will discuss recent variational and computational work on this coupling in nematic and smectic systems. In nematic droplets, simultaneous optimization of the director field and interface predicts tactoid shapes and unusual defect-induced geometries, including a symmetry-breaking instability in which reorganization of the orientational field selects a new free-boundary shape. I will also describe emerging work on smectic liquid crystals, where the constraint of nearly equally spaced layers introduces much stronger geometric frustration and gives rise to focal-conic and cusped structures. These examples provide a passive setting in which to understand how order, defects and boundaries select one another, and suggest geometric mechanisms that remain relevant when nonequilibrium stresses and activity are introduced.

Speaker: Timothy Atherton (Tufts University)

15:00 Constrained active nematics

Topological active soft matter includes fluids characterized by orientational ordering of constituting entities, ranging from anisotropic particles that extract energy from their surroundings to biological and living systems that convert chemical energy. Due to its softness, topological soft matter rarely exhibits homogeneous orientational order. Most of the frustration arising from the competing effects of chirality, anisotropic elasticity, confining geometries, surface anchoring, and external fields leads to stable and metastable point defects, disclination lines, and solitons in the orientational order-parameter field. The dynamics of ordering and defects are accompanied by flows that strongly depend on activity. The low-activity dynamics of defect structures, with increasing activity, evolve from stationary to chaotic 3D motion — active nematic turbulence [1-4]. Here we focus on thin active nematic shells in a passive nematic with free boundaries that exhibit spontaneous activity-driven swimming. The synchronization of the dynamics of two such active nematic shells entangled by a disclination in a surrounding unconfined passive nematic is demonstrated [5].

Studies were done in collaboration with groups from Ljubljana (Miha Ravnik, Simon Čopar, Žiga Kos, Nika Kralj, and Andraž Gnidovec) and ESPCI Paris (Teresa Lopez Leon with students).

[1] A. Doostmohammadi, J. Ignés-Mullol, J. M. Yeomans & F. Sagués, Active nematics, Nature Comm. 9, 3246 (2018)
[2] S. Čopar, J. Aplinc, Ž. Kos, S. Žumer, and M. Ravnik, Topology of three-dimensional active nematic turbulence confined to droplets, Physical Review X 9, 031051 (2019).
[3] N. Kralj, M. Ravnik, and Ž. Kos, Defect Line Coarsening and Refinement in Active Nematics, Phys. Rev. Lett. 130, 128101 (2023).
[4] N. Kralj, M. Ravnik, and Ž. Kos, Chirality, anisotropic viscosity and elastic anisotropy in three-dimensional active nematic turbulence, Comm. Phys. 7, 222 (2024).
[5] N. Kralj, et al., in preparation.

Speaker: Slobodan Zumer (University of Ljubljana, Faculty of Mathematics and Physics)

14:00 → 16:00 Hydrodynamic Models with Moving Contact Lines: MS-6-2

Organisers: Hans Knüpfer, Lorenzo Giacomelli, Dirk Peschka

Conveners: Dr Dirk Peschka (Weierstrass-Institut für Angewandte Analysis und Stochastik), Hans Knüpfer (University of Heidelberg), Lorenzo Giacomelli (Sapienza University of Rome)

14:30 A Phase-Field Method for Curvature Flow of Networks with Triple Junction Drag

In recent years, newly available experimental measurements of grain boundary motion in polycrystalline materials challenge the assumption of the Herring angle condition at triple junctions (a well-known force balance condition). One of the proposed explanations for the discrepancy between these experimental data and the traditional model is the presence of triple junction drag. In this talk, I will review the sharp-interface model (curvature motion of networks with triple junction drag) and present a novel phase-field approximation for it.

Speaker: Yuchuan Yang (University of Michigan)

15:00 Thermodynamically consistent discretization of the stochastic thin film equation

We begin by informally demonstrating how the gradient flow structure of the deterministic thin film equation, which encodes the balance between driving capillary forces and limiting viscous forces, can be used as a foundation for the thermodynamically consistent introduction of fluctuations. This approach is then pursued at the level of a spatial discretization of the gradient flow structure, specifically, by discretizing the energy functional and the dissipative metric tensor. This leads us to derive a stochastic differential equation that can be interpreted as an approximation of the stochastic thin film equation. Subsequently, we show that a suitable choice of the discrete mobility function results in solutions that strictly preserve positivity. The talk closes by discussing the probability of solutions to the stochastic thin film equation to approach zero.

Speaker: Benjamin Gess (TU Berlin & MPI MiS Leipzig)

15:30 A variational front-tracking method for multiphase flow with triple junctions

We present and analyze a variational front-tracking method for a sharp-interface model of multiphase flow. The fluid interfaces between different phases are represented by curve networks in two space dimensions (2d) or surface clusters in three space dimensions (3d) with triple junctions where three interfaces meet, and boundary points/lines where an interface meets a fixed planar boundary. The model is described by the incompressible Navier--Stokes equations in the bulk domains, with classical interface conditions on the fluid interfaces, and appropriate boundary conditions at the triple junctions and boundary points/lines. We propose a weak formulation for the model, which combines a parametric formulation for the evolving interfaces and an Eulerian formulation for the bulk equations. We employ an unfitted discretization of the coupled formulation to obtain a fully discrete finite element method, where the existence and uniqueness of solutions can be shown under weak assumptions. The constructed method admits an unconditional stability result in terms of the discrete energy. Furthermore, we adapt the introduced method so that an exact volume preservation for each phase can be achieved for the discrete solutions. Numerical examples for three-phase flow and four-phase flow are presented to show the robustness and accuracy of the introduced methods.

Speaker: Robert Nürnberg

14:00 → 16:00 Phase Transitions and Pattern Formation: MS-8-3

Organisers: Helmut Abels, Patrik Knopf

Conveners: Helmut Abels (Universität Regensburg), Patrik Knopf (Karlsruher Institut für Technologie (KIT))

14:00 Incompressible limit and pressure jump in the Cahn-Hilliard equation

We consider a simplified model for tumor growth: a Cahn-Hillard equation with a repulsive potential, that models the pressure inside the tissue. This potential is of the form u^{\gamma}. In the so-called incompressible limit when gamma is sent to infinity, we observe that the pressure admits a jump at the free boundary of the model. This is a joint work with Benoît Perthame and Jakub Skrzeczkowski.

Speaker: Charles Elbar (Université Claude Bernard Lyon 1)

15:00 A new approach to convergence to equilibrium for Cahn--Hilliard type equations with non-degenerate mobility

I will discuss the initial and boundary value problem for the Cahn--Hilliard equation with non-degenerate mobility and singular potential, showing that any weak solution converges to a single equilibrium under minimal assumptions, i.e., the existence of a global weak solution satisfying an energy inequality. This result also holds in the three-dimensional case, which was an open problem so far due to the lack of regularity of solutions, especially when the mobility is just a continuous function.  I will then explain the rich variety of applications of this novel approach, ranging from Cahn—Hilliard-Navier--Stokes type systems with singular potential and nondegenerate mobility, to the conserved Allen—Cahn equation and even to the nonlocal Cahn—Hilliard equation in three dimensions.

Speaker: Andrea Poiatti (University of Parma)

15:30 Sharp Interface Limit of a 3D Navier--Stokes/Allen--Cahn system

In this talk, I will report a recent work on the sharp interface limit of a Navier--Stokes/Allen--Cahn system as the interfacial thickness $\varepsilon$ tends to zero for well-prepared initial data as long as the limit system possesses a sufficiently smooth solution. We propose a systematic argument based on the linearization of the errors. The convergence results relies crucially on uniform higher-order estimates for the associated linearized Navier--Stokes/Allen--Cahn system in suitably weighted $L^2$-Sobolev spaces. Here a novel problem-adapted weight proportional to the sum of ε and the distance to the sharp interface of the limit, which gives improved and sharp estimates, is an important new ingredient. This approach can be potentially adapted to other sharp interface limits as well.

Speaker: Yadong Liu (Nanjing Normal University)

16:00 → 16:30 Coffee Break

16:30 → 17:30 Plenary Talk: Matteo Novaga

Convener: José Francisco Rodrigues (Universidade de Lisboa)

16:30 The periodic Kelvin problem

The classical Kelvin problem seeks the optimal equal-volume partition of $\mathbb R^d$ minimizing boundary surface area. In this talk, we present recent progress on periodic minimizing partitions in both isotropic and anisotropic settings.
Focusing first on two-dimensional configurations, we analyze optimal planar tilings and extend these techniques to higher dimensions, exploring connections with multi-bubble partitions, phase separation models, and periodic minimal surfaces, with emphasis on regularity, qualitative properties, and asymptotic volume regimes. Finally, we discuss the minimality of the regular truncated octahedron among all 3D parallelohedra.
The results are based on joint works with A. Cesaroni, F. Nobili and E. Paolini.

Speaker: Matteo Novaga (University of Pisa)

periodic_kelvin_fbp2026.pdf

19:00 → 21:00 Conference Dinner

The conference dinner of FBP2026 takes place at the
Restaurant "Nolle",
Georgenstr. 203,
10117 Berlin

Thursday 10 September

09:00 → 10:00 Plenary Talk: Senjo Shimizu

Convener: Hans Knüpfer (University of Heidelberg)

09:00 Free boundary problems for the Navier-Stokes equations in scaling critical spaces

A time-dependent free surface problem for the Navier-Stokes equations which describes the motion of viscous fluid in a domain close to the half-space is considered. We discuss well-posedness of the problem for an initial data in scale invariant critical Besov spaces. Our proof is based on maximal $L^1$-regularity of the corresponding Stokes problem in the half-space. Utilizing the almost orthogonal properties between the boundary potential and the Littlewood-Paley decomposition, we show maximal $L^1$-regularity in the Besov and the Lizorkin-Triebel spaces. This is a joint work with Takayoshi Ogawa (Waseda University).

Speaker: Senjo Shimizu (Kyoto University)

10:00 → 10:30 Coffee Break

10:30 → 12:30 Degenerate PDEs and Free Boundary Problems: MS-1-1

Organisers: Damiāo J. Araújo, Miguel Urbano

Convener: José Miguel Urbano (KAUST)

10:30 An unstable parabolic problem

We will discuss a parabolic PDE that arises in models for combustion and wildfire. Our main concern is the fire front: how it moves and what shape it can take. In the study of PDEs, the fire front is the free boundary. In this problem, at a fixed time the shape of the fire front can have singularities. However, when the fire front is viewed as a free boundary in space-time, we show that the free boundary is a $C^{1,\alpha}$ manifold. This is joint work with Gilles Bokolo-Tamba.

Speaker: Mark Allen (Brigham Young University)

11:00 Multiscale Mechanisms in Elliptic Regularity

Many equations arising in mathematics and the sciences have a remarkable smoothing effect. The heat equation provides a striking example: even when starting from very irregular initial data, its solutions become smooth instantaneously. Classical regularity theory seeks to explain this phenomenon for broad classes of elliptic and parabolic equations. In many important problems, however, the regularizing mechanism is only partially present. It may operate only in certain regions or above a distinguished scale, as in problems arising from homogenization and numerical schemes. In this talk, I will revisit some results from the classical theory and present recent work on multiscale Schauder estimates.

Applications include higher-order regularity for nonlocal equations with rough kernels, further time regularity for parabolic equations, free boundary problems, and problems whose elliptic structure degenerates below a scale-dependent threshold.

Speaker: Hector Chang-Lara (Universidade de Coimbra and CIMAT)

11:30 Constraint maps: singularities vs. free boundaries

Constraint maps are energy-minimizers under an image constraint, which is a natural vectorial extension of the obstacle problem. These maps develop free boundaries, due to the presence of an obstacle, while they also develop singularities, such as discontinuities, often for topological reasons. The partial regularity theory of these maps was established in the nineties, little has been known concerning their free boundaries. In this talk, I will discuss recent development on the intricate interplay between free boundaries and singularities. This talk is based on a series of joint works with Alessio Figalli (ETH), André Guerra (Cambridge), and Henrik Shahgholian (KTH), as well as recent joint work with Marvin Weidner (U. Bonn) and Hui Yu (NUS).

Speaker: Sunghan Kim (KTH Royal Institute of Technology)

12:00 A free boundary problem with non-local obstacle

Consider the cylindrical domain $\Omega=D\times(0,1)$ and the convex functional
$$ \int_\Omega \frac{1}{2}|\nabla U(x)|^2dx +\int_D V(x')^+\,dx', $$ with nonlocal obstacle acting on function $V(x')=\int_0^1 U(x', t) dt $. We show that the unique minimizer solves the equation $$ \Delta U(x',x_n) = \chi_{\{V>0\}}(x') + \chi_{\{V=0\}}(x') [\partial_\nu U (x',0) + \partial_\nu U (x',1)], $$ where $\partial_\nu U$ is the exterior normal derivative of $U$.

Several further regularity results are proven. It is shown that the comparison principle does not hold for minimizers, which makes numerical approximation somewhat challenging.

Speaker: Dr Hayk Mikayelyan (University of Nottingham Ningbo China)

10:30 → 12:30 Optimization and Free Boundaries: MS-4-1

Organisers: Amal Alphonse, Michael Hintermüller, Michael Hinze

Conveners: Amal Alphonse, Michael Hintermüller (Weierstrass Institute Berlin), Michael Hinze (Universität Koblenz)

11:30 Can we Control the Free Boundary in the Obstacle Problem?

In the classical obstacle problem, the corresponding optimal control of the variational inequality concerns usually its solution, i.e. the minimal supersolution of a Poisson equation lying above the obstacle, which is the control together with the external force. In this work we discuss the control of the free boundary in terms of different frameworks of the characteristic function of the coincidence set, which may be just a closed set, a set of finite perimeter or, in a special star-shaped geometry may have a Lipschitz free boundary.

Speaker: José Francisco Rodrigues (Universidade de Lisboa)

10:30 → 12:30 Phase Field Methods in Real-World Applications: MS-7-3

Organisers: Cecilia Cavaterra, Robert Lasarzik, Elisabetta Rocca, Hao Wu

Convener: Elisabetta Rocca (Università di Pavia)

10:30 A constrained-growth problem.

I will present some existence result for a mathematical model of accretive growth in confined environments. Accretive growth occurs ubiquitously in most biological systems, as well as in many natural and technological ones. The model describes the growth process through a level-set formulation, leading to a free-boundary problem combined with a stationary Hamilton-Jacobi equation in a prescribed constraining domain. I will discuss existence, variational characterization, and regularity properties for the growth subproblem, as well as existence for the fully coupled system obtained by combining the growth dynamics with an elliptic equation for the activation field.

Speaker: Ulisse Stefanelli (Unviersity of Vienna)

11:00 Curvature-driven pattern formation in biomembranes: A gradient flow approach

In this talk, we study a phase-field model for curvature-driven pattern formation in biomembranes. The model is derived as a gradient flow of an energy functional that approximates the two-phase Canham--Helfrich energy. This leads to a Cahn--Hilliard-type equation with cross diffusion for the relative chemical concentration of one lipid phase, coupled to a fourth-order reaction-diffusion equation describing the height profile of the membrane.
We first discuss the existence of weak solutions for the case of regular double-well potentials, using a minimizing movement scheme to construct approximate solutions. The analysis is then extended to singular potentials, e.g., the Flory--Huggins potential, by approximating them with a Moreau--Yosida regularization.
For both cases, we establish higher regularity, continuous dependence on the initial data, and consequently the uniqueness of weak solutions.
Finally, we propose a well-posed finite element discretization of the model and present numerical experiments illustrating the effect of different physical parameters on the resulting membrane patterns. Depending on the parameter regime, we observe purely striped, dotted, or snake-like structures.
(This talk is based on joint work with Patrik Knopf (UR, KIT) and Anastasija Pešić (WIAS).)

Speaker: Dennis Trautwein (Universität Regensburg)

11:30 Nonlocal-to-local convergence of convolution operators and its application to phase-field models

The goal of nonlocal-to-local convergence is to show that certain singular, nonlocal convolution-type integral operators converge to a local differential operator as the convolution kernel concentrates at zero. This can be a useful tool in the physical justification of mathematical models (e.g., the Cahn-Hilliard equation), especially when a desired local differential operator cannot be derived by microscopic laws.
The nonlocal-to-local convergence of convolution operators with radially symmetric (i.e., isotropic) kernels having $W^{1,1}$-regularity is already very well understood. However, the assumption of $W^{1,1}$-regularity is too strong for many applications. Also, in some situations (e.g., crystallization phenomena), convolution kernels are not radially symmetric but merely even (i.e., anisotropic).
In this talk, I will present very recent results from a collaboration with H. Abels and C. Hurm concerning the strong nonlocal-to-local convergence with convergence rates for anisotropic kernels, which merely need to have a significantly lower regularity than $W^{1,1}$. Also applications to Cahn--Hilliard type phase-field models will be discussed.

Speaker: Patrik Knopf (Karlsruher Institut für Technologie (KIT))

12:00 A morphoelastic phase field model predicts buckling instability in tumor growth

Growing tumors generate and respond to stress in their local environment. On the one hand, local cell divisions and death lead to complex strain patterns in the tissue. On the other hand, tissue re-arrangements can relax the resulting mechanical shear stresses and make the tissue more fluid-like. To predict the outcomes of these nonlinear visco-elastic interactions, we introduce the framework of morphoelasticity to phase field modeling of a growing tumor embedded in a surrounding host tissue.
Coupling this continuum system to diffusible growth-promoting nutrient, our simulations identify a symmetry-breaking instability in 2D and 3D, driven by two primary mechanisms: (i) elastic buckling instabiliies due to differential growth induced by the nutrient gradient and (ii) instabilities generated by apoptosis-related volumetric loss. Further, tissue fluidity and compressibility can lead to changes in tumor topologies. Our modeling framework provides a robust methodology for investigating how tissue mechanics and growth factor signaling influence the progression and invasive potential of solid tumors.

Speaker: Sebastian Aland (HTW Dresden & TU Freiberg)

10:30 → 12:30 Fluid-structure Interactions: MS-9-1

Organisers: Igor Kukavica, Yadong Liu, Sebstian Schwarzacher

Convener: Igor Kukavica (University of Southern California)

10:30 Range failure, resolvent growth, and resonance in partially dissipative systems

We consider the abstract evolution problem $u'(t)=Au(t)+f(t)$ in a
Hilbert space, where $A$ generates a strongly stable but not uniformly
stable $C_0$-semigroup $S(t)$, the typical situation for partially
dissipative systems such as coupled heat--wave models, which arise as
simplifications of fluid--structure interaction. In this setting
classical resonance, i.e.\ an eigenvalue of $A$ on the imaginary axis,
cannot occur. Nevertheless, time-periodic forcing $f$ may fail to
produce a bounded, or any, time-periodic response. We discuss several
notions of resonance appropriate to this setting, organized as a
hierarchy of failures of the range condition $\mathcal R(I-S(T))=H$,
and their connection to the growth of the resolvent of $A$ along the
imaginary axis. As the main example, we show that a heat--wave system
whose wave component occupies an elliptic domain admits periods $T$
with infinite regularity loss between forcing and solution, so that
resonance occurs already for smooth time-periodic forces.

Speaker: Boris Muha (University of Zagreb Faculty of Science)

11:00 Finite-time contact in fluid-elastic structure interactions

In this talk, we will consider a fluid-structure interaction problem involving a viscous, incompressible fluid flow, modeled by the 2D Navier-Stokes equations, through a thin deformable elastic tube, elastodynamics of which is modeled by 1D plate equations. The fluid and the structure are nonlinearly coupled at the fluid-structure interface. The fluid flow is driven by dynamic pressure data imposed at the inlet and the outlet of the tube. In this talk, we will impose the Navier-slip boundary condition at the fluid-structure interface and at the bottom rigid boundary of the fluid domain. We will first discuss the existence of weak solutions and reveal a `hidden' spatial regularity result for the structure displacement. Then we will discuss our recent result that establishes the existence of a finite time for the weak solutions at which the compliant upper boundary meets the lower boundary (i.e., the tube collapses), provided that there is a sufficient pressure drop across the channel. This resolves the ``no-collision'' paradox identified in the no-slip setting and thus validates the model to correctly capture near-contact dynamics.

Speaker: Krutika Tawri (University of Washington)

11:30 Variational methods for fluid-structure interactions

In this lecture, I will summarize how variational methods can be used to establish the existence of weak solutions for fluid–structure interaction problems. In particular, I will discuss how PDEs with inertia can be approximated variationally using gradient-flow techniques. I will focus on results for largely deforming bulk solids interacting with fluids governed by the Navier–Stokes equations, under both no-slip and Navier-slip boundary conditions. The lecture is based on joint works with B. Benesova, A. Cesik and M. Kampschulte.

Speaker: Sebastian Schwarzacher

12:30 → 14:00 Lunch Break

12:30 → 14:00 Lunch for Women*

14:00 → 16:00 Degenerate PDEs and Free Boundary Problems: MS-1-2

Organisers: Damiāo J. Araújo, Miguel Urbano

Convener: Damião J. Araujo (Universidade Federal da Paraíba)

14:00 Unified A-priori Estimates for Minimizers under p,q-Growth and Exponential Growth

In this talk, we present a unified regularity framework for variational integrals with non-uniformly elliptic integrands, including those exhibiting $p,q$-growth or exponential-type growth. We consider general energy functionals of the form
$$ \int_{\Omega} f(x, Du) \, dx, $$ where the integrand $f(x, \xi)$ may satisfy natural growth, $(p,q)$-growth, or exponential growth conditions. We establish that under suitable structural assumptions on the second derivatives of $f$ with respect to the gradient variable, any local minimizer is locally Lipschitz continuous. This key result allows us to reduce complex non-uniformly elliptic problems to a standard growth setting, where classical regularity theory can be applied. Our analysis includes models beyond the uniformly elliptic case, such as anisotropic energies, the double phase functional, the $p(x)$-Laplacian, and exponential growth integrals. We show that a-priori estimates on the gradient and second derivatives of minimizers can be derived under general conditions involving functions $g_1$, $g_2$, and $g_3$ controlling the ellipticity and the regularity of $f$. These estimates serve as a crucial step in proving higher regularity results.

The results presented extend and unify various existing regularity theories and provide new insights, particularly for variational problems where the integrand grows faster than any polynomial at infinity. We conclude with examples demonstrating the applicability of our theory in multiple settings, including degenerate, anisotropic, and exponential-type energies.

Speaker: Cintia Pacchiano (Universidad Nacional Autónoma de México)

14:30 Well-posedness for fully nonlinear free transmission problems

Fully nonlinear free transmission problems have been examined from a number of perspectives. These include the existence of solutions, regularity estimates and numerical methods. However, the uniqueness of solutions remains fairly open. We resort to a relaxed notion of viscosity solutions and relate it with the usual definition. We also impose conditions on the operators driving the free transmission problem that ensure properness and, ultimately ensure the uniqueness of relaxed solutions. We conclude by discussing some consequences of our analysis to the regularity theory of relaxed solutions.

Speaker: Edgard Pimentel (CMUC, University of Coimbra)

15:00 Dead core problems in the nonlocal setting

In this talk, we will discuss conditions for the existence of solutions exhibiting a dead core in nonlocal problems. In particular, we will analyze conditions that ensure the formation of a dead core, as well as existence arguments and examples illustrating this phenomenon in the context of nonlocal partial differential equations.

Speaker: Dr Disson dos Prazeres (Universidade Federal de Sergipe)

15:30 The degenerate quenching problem

We present recent advances on the degenerate quenching problem. Our main result establishes the local finiteness of the (n-1)-dimensional Hausdorff measure of the free boundary. The proof combines optimal gradient decay estimates, derived from an intrinsic Harnack-type inequality, with a detailed analysis of the flatness regime, where minimizers enjoy improved regularity. This approach yields an alternative proof of the classical results of Phillips and, although developed in the degenerate setting, provides insights relevant to the singular case. Furthermore, we prove that, as in the linear setting, the free boundary is rectifiable.

Speaker: Dr Rafayel Teymurazyan (KAUST)

14:00 → 16:00 Free Boundaries in Biology: MS-10-1

Organisers: Charles Elbar, Andrea Poiatti

Convener: Charles Elbar (Université Claude Bernard Lyon 1)

14:00 Local well-posedness of a model for fluidic elastic membranes

We consider a system consisting of a surface Navier-Stokes equation for incompressible fluids on an elastic inextensible membrane, whose evolution is coupled to the flow of the fluid on the surface. This leads to a highly nonlinear quasilinear geometric evolution equation of parabolic-hyperbolic type. Using a suitable parametrization we linearize the system and obtain well-posedness of it in suitable $L^2$-type Sobolev spaces. With the aid of a suitable fixed-point argument we show existence of strong solutions locally in time. This is a joint-work with Yadong Liu and Andrea Poiatti.

Speaker: Helmut Abels (Universität Regensburg)

14:30 Asymptotic problems for the Keller-Segel system with density cut-off

We study the asymptotic limits of a Keller--Segel system with porous-medium-type nonlinear diffusion and nonlinear logistic sensitivity. We identify three distinguished limits and clarify their connections with the original system. Depending on the parameter regimes, the model converges to a porous medium equation when the chemotactic sensitivity is weak and the chemical diffusion is slow, a hyperbolic Keller--Segel system when the cell diffusion vanishes, and a surface-tension-driven free-boundary problem when the chemical diffusion is slow and the attraction is strong.

Speaker: Mingyue Zhang (TU Wien)

15:00 Two-phase flows with bulk-surface interaction in an evolving domain with applications to cell biology

I will present a thermodynamically consistent model (from a collaboration with Y. Liu), 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.

Speaker: Patrik Knopf (Karlsruher Institut für Technologie (KIT))

15:30 Curvature Driven Interface Evolution: Gradient Flow Structures and Stability

The most basic examples of curvature driven interface evolution are gradient flows of the perimeter. I will discuss in this talk, at the example of Mullins-Sekerka flow, how the gradient-flow structure can be used to provide robust solution theories, bypassing the usage of comparison principles typically not available for such problems. I will focus on existence theory based on an energy dissipation principle and (modulated) stability of steepest descent in energy landscape based on an entropy-entropy dissipation structure. These are joint works with varying coauthors, consisting of Julian Fischer, Tim Laux, Theresa M. Simon, and Kerrek Stinson.

Speaker: Sebastian Hensel

14:00 → 16:00 Phase Field Methods in Real-World Applications: MS-7-4

Organisers: Cecilia Cavaterra, Robert Lasarzik, Elisabetta Rocca, Hao Wu

Convener: Robert Lasarzik (WIAS)

14:00 Well-posedness, Darcy limit, and optimal control for a Brinkman–Cahn–Hilliard phase-field system with curvature effects

We consider a phase-field model for multiphase flows in porous media coupling a Brinkman equation for the fluid motion with a generalized Cahn–Hilliard equation accounting for curvature-driven effects and mass exchange. We discuss existence results for the coupled system and, under suitable assumptions, uniqueness and continuous dependence properties. We further investigate the asymptotic transition to a Darcy-type model and establish existence of solutions for the limit problem. Finally, we address a distributed optimal control problem with the velocity field as control variable. We prove existence of optimal controls and derive first-order necessary optimality conditions by means of an adjoint system. The talk reports on a joint work with Gianni Gilardi (University of Pavia), Andrea Signori (Polytechnic of Milan), and Juergen Sprekels (WIAS Berlin and Humboldt University of Berlin).

Speaker: Prof. Pierluigi Colli (University of Pavia)

15:00 Growth of active droplets by chemical reactions: Instability and splitting

Recent studies have shown that chemical reactions in phase separating systems can lead to growth and division of droplets. The fuel from chemical reactions can lead to a sequence of growth and splitting that mimics the behavior in the transition of protocells from nonliving to living systems. Such systems where energy is externally supplied and the governing equations do not fulfil a free energy inequality are termed active systems. In this talk we provide a mathematical description based on a sharp interface Mullins-Sekerka free boundary problem that can be derived from a Cahn-Hilliard system with source terms. We are interested in analyzing the influence of certain parameters that promote growth of instabilities, leading to topology changes of perturbed stationary solutions, such as droplet splitting and merging. Accompanying numerical simulations show complex dynamical behavior, such as multiple instabilities, splitting of droplets and appearance of shell-type solutions. This is a joint work with Harald Garcke (Regensburg), Robert Nurnberg (Trento) and Andrea Signori (Milan).

Speaker: Kei Fong Lam (Hong Kong Baptist University)

15:30 A phase field model of Cahn–Hilliard type for tumour growth with mechanical effects and damage

A diffuse interface model for tumour growth in the presence of a nutrient is introduced, incorporating mechanical effects and reversible tissue damage. The highly nonlinear PDE system consists of a Cahn–Hilliard equation governing the phase separation between healthy and tumour cells, coupled to a parabolic reaction-diffusion equation for the nutrient and a hyperbolic equation for the balance of linear momentum, accounting for inertial and viscous effects. The main novelty is the presence of tissue damage, whose evolution is governed by a parabolic differential inclusion. A global-in-time existence result for weak solutions is established via a time-discretization and regularization argument. Finally, the obstructions to uniqueness are discussed.

Speaker: Giulia Cavalleri (WIAS)

14:00 → 16:00 Fluid-structure Interactions: MS-9-2

Organisers: Igor Kukavica, Yadong Liu, Sebstian Schwarzacher

Convener: Sebastian Schwarzacher

14:00 Numerical Modeling and Analysis of Self-Propelled Motion of Deformable Bodies in Fluids

Living organisms such as fish propel themselves through periodic, self-generated deformations of their body shape. Modeling this motion within a fluid-structure interaction (FSI) framework requires a careful separation of the active deformation, which is prescribed and drives the motion, from the passive, resulting motion of the body, which follows the laws of rigid body dynamics. In this talk, we present a modeling and discretization framework for such problems, building on classical monolithic ALE-based FSI approaches for fluid-rigid body and fluid-structure interaction problems.

We describe how the body's motion is decomposed into a prescribed active deformation, modeling the fish-like undulation, and an emergent rigid body motion, determined self-consistently from the interaction with the surrounding fluid. This splitting leads to a coupled system in which the rigid body position and velocity become additional unknowns, coupled monolithically to the incompressible Navier-Stokes equations through kinematic and dynamic transmission conditions on the moving boundary.

We discuss the resulting numerical discretization, based on a finite element ALE formulation, and address the solution of the strongly coupled nonlinear system arising at each time step. Since the propulsion mechanism is periodic in time, we further consider the direct computation of the time-periodic limit cycle, avoiding long transient simulations by formulating and solving a periodicity constraint as part of the coupled system.

Finally, we present first results on the optimization of the active deformation with respect to propulsion efficiency, laying the groundwork for a systematic design and control framework for self-propelled, fish-like locomotion in viscous fluids.

Speaker: Thomas Richter (Otto-von-Guericke Universität Magdeburg)

14:30 Well-posedness theorems in fluid-structure interaction: perfectly elastic shells

We consider the interaction of a 3D incompressible fluid with a 2D flexible shell that occupies (a part of) the boundary of the fluid domain. We assume that the shell is perfectly elastic while the fluid is governed by the Navier--Stokes equations. Consequently, damping within the coupled system comes entirely from the parabolic fluid subsystem. Our main result is the construction of a local-in-time unique strong solution to the system of PDEs. Standard techniques from the literature do not apply here. They are restricted to visco-elastic structures, where the corresponding solid phase is parabolic. Our construction relies on a different method built upon a new estimate for the acceleration of the system.
In the case of a 2D viscous incompressible fluid interacting with a 1D perfectly elastic shell we can extend the local solution globally in time (until a possible self-intersection of the shell).

Speaker: Dominic Breit

15:00 New results on the asymptotics of Stokes problem with application to fluid solid problems

Computing asymptotics of the Stokes problem in thin domains is a crucial issue in the analysis of fluid/solid problems. It enables a sharp description of the relative motion of particles that are close to contact and it is also a crucial step toward computing effective properties of mixtures in a dense regime. First results trace back to the book of Happel and Brenner (65’) using formal lubrication analysis. Many contributions aimed to justify rigorously these formal computations on the basis of a stability analysis or using variational arguments. In this talk I shall report on recent developments in the 3D case and in the case of domains with Lipschitz boundaries.

Speaker: Matthieu Hillairet (Université de Montpellier)

16:00 → 16:30 Coffee Break

16:30 → 17:30 Plenary Talk: Chandrasekhar Venkataraman

Convener: Axel Voigt

16:30 Free boundaries on moving boundaries - Stefan-type problems on evolving surfaces

In the first part of the talk, we consider coupled bulk-surface reaction-diffusion systems modelling receptor-ligand dynamics on an evolving domain. In biologically relevant regimes, we derive various novel free-boundary problems as limits of the model. These limiting free-boundary problems may be formulated as Stefan-type problems on an evolving hypersurface. Our results are new even in the setting where there is no domain evolution. The models are of relevance to applications in cell biology. Numerical simulations illustrating the convergence towards free-boundary problems will be presented. This part of the talk is based on joint work with Amal Alphonse (WIAS), Diogo Caetano, and Charlie Elliott (both Warwick)
In the second part of the talk, we discuss an evolving surface finite element method (ESFEM) for solving the two-phase Stefan problem on an evolving surface. In particular, we discuss the error analysis for an ESFEM discretisation of the enthalpy formulation of the two-phase Stefan problem. We will also discuss some qualitative properties of solutions to the enthalpy formulation on an evolving surface. This part is joint work with Philip Herbert (Sussex) and Tom Sales (UCL).

Speaker: Chandrasekhar Venkataraman (University of Sussex)

17:30 → 18:30 Plenary Talk: Klaus Deckelnick

Convener: Robert Nürnberg

17:30 Error analysis of a finite element scheme for parametric mean curvature flow

Mean curvature flow evolves a family of hypersurfaces in such a way that the normal velocity is given by the mean curvature at each point of the surface. This fundamental geometric evolution law can be interpreted as the $L^2$-gradient flow of the area functional and has applications e.g. in materials science and image processing. The talk is concerned with the analysis of a numerical method for the approximation of mean curvature flow in a parametric setting. Our starting point is an approach suggested by Elliott and Fritz that derives a parabolic system for the position vector of the evolving hypersurfaces using a reparametrization via the DeTurck trick. This technique introduces a tangential velocity that leads to strict parabolicity of the system. Based on a natural weak formulation we introduce a finite element semidiscretization with continuous, piecewise polynomial elements of order $k \geq 2$ that are defined on a fixed reference hypersurface. As our main result we shall present an error analysis for the resulting scheme that yields in particular an optimal $H^1$-error bound for the position vector. In addition, we show a couple of numerical results in order to illustrate that the tangential motion leads to good mesh properties. This is work obtained in collaboration with Vanessa Styles (University of Sussex).

Speaker: Klaus Deckelnick (Otto-von-Guericke-Universitaet Magdeburg)

Friday 11 September

09:00 → 11:00 Degenerate PDEs and Free Boundary Problems: MS-1-3

Organisers: Damiāo J. Araújo, Miguel Urbano

Convener: José Miguel Urbano (KAUST)

09:00 On the regularity of minimizers arising from singular minimal surface-like energies

We develop a unified variational framework for singular energies that arise as analytic models in Simon's symmetric minimal surface program. Motivated by the linearized profile $\Delta u \sim u^{-1}$, we study local minimizers of energies whose potential $\sigma$ may diverge at the origin. This class encompasses both the logarithmic potentials linked to the geometry of minimal cones and the negative-power potentials appearing in modern free boundary models.

Speaker: Aelson Sobral (King Abdullah University of Science and Technology)

10:00 Higher regularity in nonlocal free boundary problems

A classical result by Kinderlehrer-Nirenberg states that C^{1,\alpha} regularity of the free boundary near a regular point implies that the free boundary is smooth. This principle applies to a wide range of free boundary problems for second-order operators. The goal of this talk is to explore counterparts of such higher regularity results for nonlocal free boundary problems, focusing on the nonlocal one-phase and obstacle problem.

Speaker: Marvin Weidner (University of Bonn)

09:00 → 11:00 Free Boundaries in Biology: MS-10-2

Organisers: Charles Elbar, Andrea Poiatti

Convener: Andrea Poiatti (University of Parma)

09:00 The elastica functional as the critical Gamma-limit of the screened Gamow model

I will present an example for how to pass from a first order perimeter optimization problem to a second order curvature optimization problem. While the original model, a nonlocal isoperimetric problem, arises from mathematical physics, its limit is Euler’s elastica functional. This is the 2D analogue of the Willmore energy, which features prominently in the modeling of lipid bilayers. Therefore, the methods we developed are also of interest in mathematical biology.

This is joint work with Cyrill Muratov and Matteo Novaga.

Speaker: Theresa Simon (University of Münster)

09:30 On a viscoelastic Mullins-Sekerka system

In recent years, interface models have become increasingly popular in mathematical biology for describing tumour growth. Among these, the Cahn–Hilliard–Biot model was the first to incorporate both mechanical stress and fluid flow through a heterogeneous, saturated porous medium. In subsequent works, the original model was further extended to include Kelvin–Voigt viscoelasticity. Since the Cahn–Hilliard–Biot system is a diffuse-interface model, it is natural to ask about related sharp-interface systems.
Motivated by this question, we introduce a novel viscoelastic Mullins–Sekerka system with second-gradient terms and a constant contact angle at the boundary, which we derive as an H⁻¹-H¹-type gradient flow of the perimeter and an elastic energy. Moreover, we discuss different notions of weak solutions, namely BV solutions and varifold solutions satisfying an optimal energy dissipation principle in the spirit of De Giorgi.
This is joint work with Helmut Abels and Harald Garcke (Universität Regensburg)

Speaker: Jonas Haselböck (Universität Regensburg)

10:00 Interface dynamics in viscoelastic phase separation

Viscoelastic phase separation is thought to play an important role in cell biology. Differences in mechanical properties and relaxation times of the constituents can induce dynamic asymmetry during demixing, generating complex morphological transitions such as phase inversion, a characteristic feature of viscoelastic phase separation. In this talk, we determine the interface dynamics arising from formal sharp-interface asymptotics in a degenerate Cahn–Hilliard model for viscoelastic phase separation with cross-diffusive coupling to a bulk stress variable. We obtain non-local, lower-order counterparts of the classical surface diffusion flow: for a constant coupling function, we recover the intermediate surface diffusion flow, whereas phase-dependent coupling leads to a class of fractional interface laws whose propagation operator, at principal order, is the square root of the minus Laplace–Beltrami operator.

Speaker: Katharina Hopf

10:30 On Cahn-Hilliard-Keller-Segel type models in biology

In this talk we introduce a mathematical model which couples the evolution of a phase-parameter $\varphi$ satisfying a Cahn-Hilliard type relation with the one of an additional variable $\sigma$ influencing the phase separation process. The main application of the model refers to cancer growth processes, where $\sigma$ may represent the concentration of a chemical substance affecting the evolution of the tumor, and is governed by a nonlinear parabolic equation characterized by a cross-diffusion term alike that occurring in the Keller-Segel model for chemotaxis. This term is also responsible for the most relevant difficulties in the mathematical analysis of the system.
Complementing previous results on the model, we show global in time existence for a very weak notion of solution to which a suitable energy imbalance and a logarithmic inequality for the nutrient are added. Noting that the system also admits local
in time ``strong'' solutions, we can also exhibit a weak-strong uniqueness result whose proof exploits in an essential way the entropy-type inequality satisfied by weak solutions.

Speaker: Elisabetta Rocca (Università di Pavia)

09:00 → 11:00 Optimization and Free Boundaries: MS-4-2

Organisers: Amal Alphonse, Michael Hintermüller, Michael Hinze

Conveners: Amal Alphonse, Michael Hintermüller (Weierstrass Institute Berlin), Michael Hinze (Universität Koblenz)

09:00 Optimal design of linearly Cosserat and thermal spectral structures: A phase field approach

We study a phase-field structural topology optimization problem motivated by applications in 3D printing formulated using a thermoelastic Cosserat continuum framework with thermal spectral dissipation. The Cosserat theory accounts for micropolar effects in the mechanical response, while thermal effects account for heat-driven phenomena relevant to additive manufacturing. The resulting optimization problem includes an anisotropic Ginzburg-Landau functional in order to produce optimal designs that adhere to the overhang angle constraints that plays a significant role in determining the success of the printing process. We present results concerning existence of minimizers and derive first order necessary optimality conditions using subdifferential calculus to handle the non-differentiable anisotropy term. If time permits, we will discuss related sharp interface limits and second order sufficient optimality conditions under simplified settings. This is a joint work with Ting Fung Chan (Hong Kong)

Speaker: Kei Fong Lam (Hong Kong Baptist University)

09:30 Convergence of a shape optimisation algorithm for a semi-linear elliptic equtions in function spaces using $W^{1,\infty}$-steepest descent

In this talk, we consider the shape optimisation of a semi-linear elliptic equation. We use a $W^{1,\infty}$-steepest descent with Armio step size to perform the optimisation. In the infinite dimensional setting, we show the convergence of two-dimensional shapes under a mild assumption. We conclude with some numerical experiments.

Speaker: Philip Herbert (University of Sussex)

10:00 Reduced Order CFD based Reactor Shape Optimization for Heterogeneous Catalytic Systems

Computational Fluid Dynamics (CFD) analysis of reactive flows over heterogeneous catalysts is a challenging task even for “simple” laminar flows. This involves solving a system of Partial Differential Equations (PDEs) with highly non-linear boundary conditions imposed by the surface chemistry which leads to stiffness and bad conditioning of the overall equation system requiring extensive computational effort. To address this challenge, we recently developed reduced order models (ROMs) for heterogeneous catalytic systems (a,b). These ROMs have two advantages: (1) Their pre-processing or offline part is cheaper than other model order reduction methods or machine learning approaches as it does not require solving the high-fidelity simulations and (2) the online part requires only solving non-linear equations (NLEs) rather than a partial differential equation (PDE) system. Since the ROMs enable solving the governing non-linear PDE system with minimal computational effort, it becomes feasible to couple them directly with an optimizer as opposed to full scale CFD. Therefore, one can think of extending these ROMs to applications such as reactor design optimization. In the context of heterogeneous catalysis, this could revolutionize how catalytic reactors are designed especially with the onset of 3D printing techniques.
To this end, we couple the ROMs with the shape optimization toolbox Fireshape (c), which implements a moving-mesh approach to update the computational domain and minimize a target function, which in our case could be the product yield or selectivity. Fireshape has already been used effectively for optimizing geometries where the flow is described by incompressible Navier Stokes.
The complete workflow for optimal reactor design is as follows: The optimizer passes the initial shape to the finite element solver – Firedrake (d) for computation of the non-reactive part of flow comprising the incompressible Navier Stokes equations. The transport properties and velocity fields are computed in this step. Then for the same shape, snapshots (which are PDE systems with piecewise constant Neumann boundary conditions) are generated using Firedrake. The reactive part of the solution is assumed to be a linear combination of these snapshots. The coefficients for each snapshot are then computed via a non-linear algebraic equations’ solver. The non-reactive and reactive solutions are combined to estimate the ROM solution. This solution is then passed back to the optimizer which computes the next iterate (shape) based on the gradients computed via automatic differentiation. In this work, we apply this reactor design workflow on different kinetic systems for catalytic monolith geometries.

References:
a) Muhammad Uzair Qureshi et al. Reduced order CFD modeling approach based on the asymptotic expansion An application for heterogeneous catalytic systems : Chemical Engineering Journal 504 (Jan. 2025), p. 158684.

b) S. Matera, C. Merdon, and D. Runge. Reduced Basis Approach for Convection-Diffusion Equations with Non-linear Boundary Reaction Conditions””. In: Finite Volumes for Complex Applications X Volume 1, Elliptic and Parabolic Problems. Springer Nature Switzerland, 2023, pp. 335 343.

c) Alberto Paganini and Florian Wechsung. Fireshape: a shape optimization toolbox for Firedrake””. In: Structural and Multidisciplinary Optimization 63.5 (Feb. 2021), pp. 2553 2569.

d) Ham, David A., et al. Firedrake User Manual. First ed., Imperial College London; University of Oxford; Baylor University; University of Washington, May 2023.

Speaker: Muhammad Uzair Qureshi

09:00 → 11:00 Fluid-structure Interactions: MS-9-3

Organisers: Igor Kukavica, Yadong Liu, Sebstian Schwarzacher

Convener: Yadong Liu (Nanjing Normal University)

09:00 On the motion of a rigid body in a perfect compressible fluid

We study the interaction between a rigid body and an inviscid compressible fluid in a bounded domain. The fluid is described by the compressible Euler equations, while the rigid body obeys the laws of linear and angular momentum. We prove the existence and uniqueness of classical solutions to the resulting coupled fluid–structure interaction system. The analysis relies on a suitable approximation procedure combined with uniform estimates and compactness arguments.

Speaker: Pei SU

10:00 Collective effect of a large system of heavy particles immersed in a Newtonian fluid

We consider the motion of a large number of heavy particles in
a Newtonian fluid occupying a bounded spatial domain. When we say ``heavy", we mean a particle with a mass density that approaches infinity at an appropriate rate as its radius vanishes. We show that the collective effect of heavy particles on the fluid motion is similar to the Brinkman perturbation of the Navier-Stokes system identified in the homogenization process.

Speaker: Prof. Arnab Roy (Basque Center for Applied Mathematics)

10:30 Long-Time Behavior of Fluid-Structure Interaction Models

We address a system of partial differential equations modeling the motion of an elastic body inside an incompressible fluid. The fluid is governed by the incompressible Navier-Stokes equations, while the structure is represented by the wave equation. In this talk, we review established local and global existence theorems and discuss recent developments in their long-time dynamics. These results are based on joint works with M. Ignatova, B. Ingimarson, I. Lasiecka, W. Ożański, and A. Tuffaha.

Speaker: Igor Kukavica (University of Southern California)

11:00 → 11:30 Coffee Break

11:30 → 12:30 Plenary Talk: Balázs Kovács

Convener: Buyang Li (The Hong Kong Polytechnic University)

11:30 A Lagrange-multiplier algorithm for mean curvature flow: structure-preservation and error estimates

In this talk we will present and discuss a fully discrete algorithm for mean curvature flow of closed surfaces using Lagrange multipliers for preserving the energy-decreasing structure. The algorithm is based on the solution-driven formulation using Huisken's evolution equations for the normal vector and the mean curvature. The method uses a high-order multistep methods in time and evolving surface finite elements in space.
The approach also accommodates artificial tangential velocities of minimal deformation rate-type (MDR-type). The resulting fully discrete algorithms are area-decreasing at every time step, with a prescribed decay rate determined by the computed mean curvature.

We will discuss local existence and uniqueness of the discrete Lagrange multiplier and convergence of a simplified Newton iteration for its computation under weak regularity assumptions, and present optimal-order error estimates without and with tangential motion.

The talk is based on joint work with Christian Lubich (Tübingen) and Buyang Li (PolyU Hong Kong).

Speaker: Balázs Kovács (Paderborn University)

12:30 → 13:30 Plenary Talk: Maxim Olshanskii

Convener: Michael Hinze (Universität Koblenz)

12:30 Parabolic PDEs on evolving domains with topological transitions: well-posedness and numerical analysis

The talk concerns linear parabolic problems posed on time-dependent domains that may undergo topological changes. The domains are represented by smooth level-set functions and may split, merge, or develop or lose components and holes. Using the classification of generic transitions provided by Morse theory, we introduce anisotropic space-time function spaces adapted to the resulting singular geometries. This framework yields existence, uniqueness, and a priori estimates for weak solutions.

The talk then focuses on an unfitted finite element method for such and on its error analysis. Several structural assumptions on the domain evolution near the critical time provide control over the variation of solution norms, even across the singularity, and form the foundation of the stability and error analysis. Their applicability is illustrated by level-set domains undergoing different topological transitions. Optimal-order error estimates are supported by numerical experiments, and limitations of the current analysis and open questions will be discussed.

Speaker: Maxim Olshanskii (University of Houston)

13:30 → 13:45 Plenary Talk: Closing Remarks

13:45 → 14:45 Lunch Break and End of Conference