In this talk we introduce a mathematical model which couples the evolution of a phase-parameter satisfying a Cahn-Hilliard type relation with the one of an additional variable influencing the phase separation process. The main application of the model refers to cancer growth processes, where the concentration of a chemical substance, affecting the evolution of the tumor, 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.
The content of the talk is contained in two joint works with Robert Lasarzik, Giulio Schimperna, and Andrea Signori.

The Cahn–Hilliard equation is one of the most widely used models for describing phase separation processes and plays an important role in the mathematical modeling of two-phase flows.

Despite having been studied extensively in the literature, the classical (local) Cahn–Hilliard equation has the limitation that the underlying free energy functional lacks a direct physical derivation from microscopic principles. To overcome this issue, Giacomin and Lebowitz derived a nonlocal variant of the Cahn–Hilliard equation. We discuss how local Cahn–Hilliard models can be justified through their nonlocal counterparts by means of nonlocal-to-local convergence.

A further limitation of standard Cahn–Hilliard models is the use of homogeneous Neumann boundary conditions for both the phase-field variable and the chemical potential. These boundary conditions impose two fundamental restrictions:

1. The diffuse interface separating the two phases is forced to meet the boundary at a right angle, which is unrealistic in many applications.

2. No mass transfer between the bulk and the boundary is permitted, and hence absorption processes cannot be described.

To overcome these limitations, Cahn–Hilliard equations with dynamic boundary conditions have been introduced. We study dynamic boundary conditions that themselves exhibit a Cahn–Hilliard-type structure and discuss related Navier–Stokes–Cahn–Hilliard models for two-phase flows.

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

The work presented in the talk is based on a joint collaboration with Andrea Di Primio (Scuola Normale Superiore, Pisa, Italy) and Andrea Papini (University of Gothenburg, Sweden).

The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening takes place and, in contrast, growth and division of droplets occur, none of which grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. The sharp interface limit is a nonlocal geometric evolution equation of Mullins–Sekerka type. We will study the stability of stationary solutions and identify parameters that lead to instabilities. We show that these instabilities can lead to topology changes which can result in the splitting of so-called protocells. We then introduce an unfitted finite-element method for the modified Mullins–Sekerka probem. In addition, we present numerical simulations that will show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures. This supports claims in biophysics which state that featureless aggregates of abiotic matter may evolve and form protocells, which can be the basis for systems that gain the structure and functions necessary to fulfill the criteria of life.

We analyze a stochastic Allen-Cahn equation for the dynamics of biomolecular damage and repair. The system is driven by two distinct noise processes: a multiplicative cylindrical Wiener process, modeling continuous background stochastic fluctuations, and a jump-type noise, modeling the abrupt, localized damage induced by external shocks. The drift of the equation is singular and covers the typical logarithmic Flory-Huggins potential required in phase-separation dynamics. We prove well-posedness of the model in a strong probabilistic sense, and analyze its long-time behavior in terms of existence and uniqueness of invariant measures, ergodicity, and mixing properties.

The talk is based on a joint work with A. Di Primio, M. Fritz and L. Scarpa.

The Cahn-Hilliard-Biot system is a diffuse-interface model describing the flow of a fluid through a deformable porous medium consisting of two phases, with applications ranging from geology to tumour growth modelling. The system nonlinearly couples Biot’s equations for poroelasticity, including phase-field-dependent material properties, with the Cahn-Hilliard equation governing the evolution of the solid. We further distinguish between the absence or presence of a viscoelastic term of Kelvin-Voigt type.

In this talk, we present recent results on the well-posedness of the system. In particular, exploiting its intrinsic gradient-flow structure, we establish global existence of weak solutions and further prove local-in-time well-posedness by means of maximal regularity theory.

These results are based on joint work with Helmut Abels and Harald Garcke.

We study an elastic compound that is exposed to rate-independent delamination processes along prescribed interfaces. The interfaces are described by convex polyhedra of random geometry. By means of stochastic two-scale homogenization we deduce a macroscopic phase-field model for rate-independent volume damage.

This is joint work in progress with Martin Heida (WIAS) and Leon Schütz, supported by the German Research Foundation (DFG) within CRC 1114 Scaling Cascades in Complex Systems, project B09 Materials with Discontinuities on Many Scales and within CRC/TRR 388 Rough Analysis, Stochastic Dynamics & Related fields, project B09 Mean field theories and scaling limits of nonlinear stochastic evolution systems.

In this talk, we present an existence result for a stochastic model describing the dynamics of collective behaviour of oxygen-driven swimming bacteria in an aquatic fluid flowing in a bounded domain, subject to random external forces. The model under consideration consists of the stochastic Navier-Stokes equations coupled with Keller-Segel equations.

Our aim in this talk is to discuss the Cahn-Hilliard equation with a (nonlinear) source term and a logarithmic nonlinear term. Such an equation has applications in, e.g., tumor growth and image processing. In particular, we discuss the existence of weak solutions and their regularity

• t.b.a.

A topology optimization problem in a phase field setting is considered to obtain rigid structures, which are resilient to external forces and constructable with additive manufacturing. Hence, large deformations of unsupported overhangs due to gravity shall be avoided during construction. The deformations depend on the stage of the construction and are modelled by linear elasticity equations on growing domains with height-dependent stress tensors and forces. Herewith, possible hardening effects can be included. The height continuous as well as in the height discretized -so called layer-by layer model is considered. Analytical results concerning the existence of minimizers and the differentiability of the reduced cost functional are presented. A key ingredient is to show, that Korn’s inequality holds with a constant independent of the height.

The problem is numerically solved using a projected gradient type method in function space. Second order information can be included by changing the underlying inner product in every iteration. We also mention some details on the implementation for additional speed up.Convergence in function space indicates that the iteration numbers are bounded independently of the discretization level and of the number of construction layers. Numerical evidence of this and the efficiency of the method are illustrated. Furthermore, the choice of weights for the penalization of deformation during the construction is discussed. For various manufacturing problem settings, results are presented for one or two materials and void in two as well as in three dimensions.

This is a joint work with Maximilian Urmann.

• t.b.a.

The numerical simulation of viscoelastic two-phase flows involves complex interface dynamics and faces major challenges, such as the High Weissenberg Number Problem and the loss of positive definiteness of the viscoelastic 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 single-phase viscoelastic models and highlight novel convergence results. Second, we extend these approaches to the two-phase setting by employing a phase-field method for the coupled fluid-interface system. Addressing the main analytical challenges, we integrate these concepts to guarantee both exact volume conservation and energy dissipation at the discrete level. Finally, we demonstrate the robustness of the proposed methods with numerical experiments.

• joint work with E. Hausenblas (Montanuniversität Leoben) and J. M. Tölle (Aalto University)

One standard way to prove existence for deterministic, highly nonlinear PDEs is to use the Schauder–Tychonoff fixed-point theorem. In what follows, we introduce and verify a stochastic variant of the Schauder–Tychonoff theorem. We apply our existence result to nonlinear stochastic diffusion equations with non-Lipschitz perturbations.

References:
[1] E. Hausenblas, A. Kumar and J. M. Tölle, A stochastic Schauder-Tychonoff type theorem and its applications, Submitted, (2026), https://arxiv.org/pdf/2602.17285.
[2] E. Hausenblas and J. M. Tölle, The stochastic Klausmeier system and a stochastic Schauder-Tychonoff type theorem. Potential Anal., 61(2):185–246, 2024.

• t.b.a.

Phase-field models involving nonsmooth terms arise in several contexts. This e.g. includes bound or simplex constraints for binary and multi-phase separation or transition problems with obstacle-type or singular potentials, non-healing conditions for phase-field fracture models, and nonsmooth dissipation potentials for ductile fracture models. In all these cases the non-smoothness prohibits the application of classical nonlinear methods and thus makes the numerical treatment particularly challenging. We present a class of algorithms that exploit the variational structure and combine nonsmooth optimization and multilevel techniques leading to robust and highly efficient iterative methods for such problems.