I will present results on the existence and uniqueness of solutions when a parabolic PDE interacts with a hyperbolic PDE. In this setting geometric conditions will be explored that allow to show the existence of unique solutions.  The regularity estimates of the solutions are sensitive to the geometry: Two different geometric conditions result in a weaker and stronger regularity estimate. Optimality of them will also be discussed. The work presented was established in collaboration with S. Mosny, B. Muha and J. Webster.

I will talk about the existence of variational solutions to doubly nonlinear parabolic PDEs in noncylindrical domains $E \subset \mathbb{R}^n \times [0,\infty)$. This setting arises from models where the underlying domain $E^t := \{ x \in \mathbb{R}^n : (x,t) \in E \}$ changes in time. The prototype of the considered PDEs is $$ \partial_t \big( |u|^{q-1} u \big) - \operatorname{div}\big( |Du|^{p-2} Du \big) = 0 \quad\text{in } E $$ with parameters $q \in (0,\infty)$ and $p \in (1,\infty)$, which combines the porous medium equation and the parabolic $p$-Laplacian. The talk is based on joint work (in progress) with Christoph Scheven, Jarkko Siltakoski and Calvin Stanko.

In this talk we focus on the nonlocal-to-local asymptotics for a tumor-growth model coupling a viscous Cahn-Hilliard equation describing the tumor proportion with a reaction-diffusion equation for the nutrient phase parameter. First, we prove that solutions to the nonlocal Cahn-Hilliard system converge, as the nonlocality parameter tends to zero, to solutions to its local counterpart. Second, we provide first-order optimality conditions for an optimal control problem on the local model, accounting also for chemotaxis, and both for regular or singular potentials, without any additional regularity assumptions on the solution operator. This is a joined work obtained in collaboration with Elisa Davoli, Elisabetta Rocca and Luca Scarpa.

The "index of hypocoercivity" is defined via a coercivity-type estimate for the self-adjoint/skew-adjoint parts of the generator, and it quantifies `how degenerate' a hypocoercive evolution equation is, both for ODEs and for evolutions equations in a Hilbert space. We show that this index characterizes the polynomial decay of the propagator norm for short time and illustrate these concepts for the Lorentz kinetic equation on a torus. Discrete time analogues of the above systems (obtained via the mid-point rule) are contractive, but typically not strictly contractive. For this setting we introduce "hypocontractivity" and an "index of hypocontractivity" and discuss their close connection to the continuous time evolution equations. This talk is based on joint work with F. Achleitner, E. Carlen, E. Nigsch, and V. Mehrmann. References: 1) F. Achleitner, A. Arnold, E. Carlen, The Hypocoercivity Index for the short time behavior of linear time-invariant ODE systems, J. of Differential Equations (2023). 2) A. Arnold, B. Signorello, Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium, Kinetic and Related Models (2022). 3) F. Achleitner, A. Arnold, V. Mehrmann, E. Nigsch, Hypocoercivity in Hilbert spaces, J. of Functional Analysis (2025).

Lower semicontinuity of surface energies in integral form is known to be equivalent to BV-ellipticity of the surface density. In this talk, we prove that BV-ellipticity coincides with the simpler notion of biconvexity for a class of densities that depend only on the jump height and jump normal, and are positively 1-homogeneous in the first argument. The second main result is the analogous statement in the setting of bounded deformations, where we show that BD-ellipticity reduces to symmetric biconvexity. Our techniques are primarily inspired by constructions from the analysis of structured deformations and the general theory of free discontinuity problems. This is joint work with Carolin Kreisbeck and Marco Morandotti.

We investigate the optimal shape of two sets of given volume which are minimizing the $\ell_1$ double-bubble type energy, i.e., the length of the outer perimeter of the configuration plus the length of the interface between the two sets, calculated with respect to the $\ell_1$ norm. We study this problem in the general case for sets of finite perimeter without any geometric constraints on their structure. We address in two dimensions for all possible volume ratios (and all possible interaction intensities) and in three dimensions for a range of volume ratios between 1/2 and 2. Our main result is the complete classification of minimisers. The talk is based on a joint work with M. Friedrich and U. Stefanelli.

In extremely thin ferromagnetic films, an additional interaction, the so-called Dzyaloshinskii-Moriya interaction (DMI), arises in the micromagnetic energy. In such materials, topoligically nontrivial, point-like configurations of the magnetization called magnetic skyrmions are observed, which are of great interest in the physics community due to possible applications in high-density data storage. We characterize skyrmions as (potentially only local) minimizers of a micromagnetic energy augmented by DMI on bounded domains and describe their asymptotics in the regime of dominating exchange (or Dirichlet) energy with an emphasis on the effect of boundary values. In the case of Dirichlet boundary data, we will also discuss the question of higher degree minimizers.

In this talk, I present the formation of singularities in a baby Skyrme type energy model, which describes magnetic solitons in two-dimensional ferromagnetic systems. In presence of a diverging anisotropy term, which enforces a preferred background state of the magnetization, I show how to establish a weak compactness result of the topological charge density and show that it converges to an atomic measure with quantized weights. I characterize the $\Gamma$-limit of the energies as the total variation of this measure. In the case of lattice type energies, I first need to carefully define a notion of discrete topological charge for $\mathbb{S}^2$-valued maps. I then prove a corresponding compactness and $\Gamma$-convergence result, thereby bridging the discrete and continuum theories.

I will discuss evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on the case where the dynamics are driven by an entropy functional defined both on the metric edges and vertices, we provide a rigorous understanding of such coupled systems of ordinary and partial differential equations as (generalized) gradient flows in continuity equation format. Approximating the edges by a sequence of vertices, which yields a fully discrete system, we are able to establish existence of solutions in this formalism. Furthermore, we study several scaling limits using the recently developed framework of EDP convergence with embeddings to rigorously show convergence to gradient flows on reduced metric and combinatorial graphs.  Finally, numerical studies confirm our theoretical findings and provide additional insights into the dynamics under rescaling. This is joint work with Jan-Frederik Pietschmann and André Schlichting.

It has been proposed that the density fluctuations of a system of weakly interacting particles in the regime of large but finite particle number are captured by a super-critical singular SPDE -the Dean-Kawasaki equation. Due to the ever expanding use of the Dean-Kawasaki equation in applications, there is a desire to rigorously justify it. In this talk we show that, using a suitable weak distance, the law of the fluctuations as predicted by a spatially discretized Dean-Kawasaki equation approximates the law of the fluctuations of the particle system up to a term that is of arbitrary order in the inverse particle number and a numerical error. Time permitting we shall also discuss how, under more restrictive conditions, cross-diffusion effects may be taken into account. This talk is based on a joint works with Federico Cornalba, Julian Fischer, and Jonas Ingmanns.

Over the past three decades, a variety of energy functionals have been developed and analyzed to address the problem of finding optimal embeddings of curves, links, and higher-dimensional submanifolds into Euclidean space. The associated Euler-Lagrange equations give rise to a rich class of geometric non-local differential equations, encompassing degenerate elliptic, singular, critical, and subcritical types. In this talk, we will provide a concise overview of the state of the art in this field, highlighting key results, challenges, and open questions.

The (dual) semi-geostrophic equations are a nonlinear coupled transport/Monge-Ampere system modeling large-scale atmospheric dynamics. G. Loeper showed in 2006 that in an appropriate scaling they can be seen as a fully nonlinear perturbation to the 2D Euler vorticity equations, a nonlinear coupled transport/Poisson system. The error between original and approximate solutions is shown to be of order $O(\epsilon)$. In this talk we present a generalization of this leading order approximation to higher-order correction terms that are obtained by an hierarchy of linear inhomogeneous coupled transport/Poisson systems and show that the error between a solution of the original system and the expansion provided by the approximate equations is of the order $O(\epsilon^n)$. This is joint work with V. Kalivopoulos, supported by the Hellenic Foundation for Research and Innovation (HFRI).