Abstracts

Helmut Abels (Universität Regensburg)
Diffuse Interface Models for Two-Phase Flows with Different Densities
Abstract: We review different diffuse interface models for the two-phase flow of two macro-scopically immiscible, incompressible fluids in the case when the densities of the fluids are different. In such models a partial mixing of the fluids in an interfacial of small thickness is taken into account. We discuss the derivation and mathematical analysis of such models. In particular, we present recent results for the model proposed by Abels, Garcke and Grün in 2012. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as time tends to infinity.
This is a joint work with Harald Garcke and Andrea Giorgini.
Michiel Bertsch (Università degli Studi di Roma Tor Vergata)
Multiscale reaction-diffusion equations on networks
Abstract: Macroscopic modelling of Alzheimer’s Disease (AD) naturally leads to equations on networks which describe the evolution of toxic proteins in the brain, such as tau and beta-amyloid. For example, the system of neurons in the brain can be easily identified with a graph. In AD, two natural but quite different timescales show up: a short one (seconds, minutes, hours…) to describe most of the involved microscopic processes such as the diffusion, aggregation and fragmentation of proteins, and a long one (years) to describe the evolution of the disease itself. One of the goals of our research is to identify possible mechanisms to explain „why in AD the long timescale is so extremely long“, possibly in terms of underlying microscopic processes. A well-known approach to multiscale problems is that of quasi-static modelling, where the short timescale becomes instantaneous in the long timescale. I will discuss some mathematical issues related to quasi-static limits.
Joint work with Emilia Cozzolino, Veronica Tora (Rome) and Nuutti Barron, Ashish Raj, Justin Torok (UCSF).
Nicolas Dirr (Cardiff University)
On the choice of multiplicative noise for reversibility
Abstract: We study the interplay between reversibility, geometry, and choice of the (in particular Ito, Stratonovich, Klimontovich) noise in stochastic differential equations (SDEs) with multiplicative noise. Building on a unified geometric framework, we derive algebraic conditions under which a diffusion process is reversible with respect to a Gibbs measure on a Riemannian manifold. For reversible slow-fast systems of SDEs with a block-diagonal diffusion structure, we show, using the theory of Dirichlet forms, that both reversibility and the Klimontovich noise interpretation are preserved under coarse-graining.
Joint work with Mario Ayala, Gigorios Pavliotis and Johannes Zimmer
Julian Fischer (Institute of Science and Technology Austria)
Hölder regularity for weak solutions to the thin-film equation in 2d
The thin-film equation ∂_tu = − ∇ ∙(uⁿ∇ ∆ u) describes the evolution of the height u = u(x, t) ≥ 0 of a viscous thin liquid film spreading on a flat solid surface. While the existence theory for weak solutions to the thin-film equation was developed more than two decades ago, much less has been known about its regularity theory: In d > 1 spatial dimensions, even boundedness of weak solutions has remained a major unsolved problem in the theory of the thin-film equation. Due the fourth-order structure of the thin-film equation, comparison principles or De Giorgi-Nash-Moser theory are not applicable.
In the physically most relevant case of two spatial dimensions d = 2, we prove Hölder continuity for energy-dissipating weak solutions to the thin-film equation as constructed by Grün. Our proof is based on a hole-filling principle, the main challenge being posed by the degenerate parabolicity of the PDE.
To address the possibly degenerate parabolicity, we crucially make use of the Bernis-Grün inequalities.
Marco Fontelos (Instituto de Ciencias Matemáticas, Madrid)
Bifurcation diagrams of singularities in PDEs
Abstract: We review the role of bifurcation theory in the analysis of singular solutions to various PDEs. We start with a nonlocal transport equation where branches of singular solutions can be continued starting from solutions to Burgers equation. We also discuss a coagulation model and a thin film model where branches of discretely selfsimilar solutions emerge.
Lorenzo Giacomelli (Sapienza Università di Roma)
Evolution of support for the thin-film equation with linear diffusion
Abstract: The analysis of qualitative properties of solutions, such as finite speed of propagation and waiting time, is among Guenther Gruen’s most prominent contributions to the theory of thin-film equations (and more). In this framework, I will discuss an ongoing joint project with Joshua Utley and him, focusing on the competition between the thin-film operator and the Laplacian in determining the support’s evolution.
Stefan Metzger (FAU Erlangen-Nürnberg)
An augmented SAV scheme for the stochastic phase-field equations
The scalar auxiliary variable (SAV) method was originally introduced for the discretization of deterministic gradient flows. By introducing an additional scalar auxiliary variable, this method allows to formulate linear numerical schemes that are still unconditionally stable with respect to a modified energy.
This talk addresses the extension of the SAV method to nonlinear stochastic partial differential equations with multiplicative noise. Using the stochastic Allen-Cahn equation as a prototype problem, we motivate why a straightforward application of the SAV method will in general not provide satisfactory results and present an augmented SAV method that remedies the shortcomings and allows for a rigorous convergence proof. We conclude by discussing the applicability of this approach to a stochastic phase-field model with stochastic dynamic boundary conditions describing contact line tension effects.
Barbara Niethammer (Universität Bonn)
Gelation versus mass conservation in a spatially inhomogeneous coagulation equation
Abstract: The classical coagulation equation, derived by Smoluchowski in 1916, models mass aggregation in various applications such as aerosol physics, polymerization, population dynamics, or astrophysics. A fundamental property is gelation, that is solutions loose mass in finite time. It is well known that this typically happens if the coagulation kernel has homogeneity larger than one.
In this talk we discuss a spatially inhomogeneous coagulation model that can describe, for example, the formation of rain droplets. It takes the effect of gravity through an additional transport term in the spatial variable into account. We establish that solutions to the homogeneous version of the equation loose mass instantaneously in time, while in contrast, solutions to the spatially inhomogeneous model conserve the mass, at least for short times.
(Joint work with Iulia Cristian and Juan Velázquez)
Andreas Prohl (Universität Tübingen)
Convergent Discretization of the Fokker-Planck equation in Higher Dimensions
Abstract: I present a new algorithm that combines the Euler scheme to solve the associated SDE with a data-dependent density estimator. I will discuss its convergence proof, which also employs tools from statistics.