Research

My research interests lie in the field of partial differential equations and applied analysis.  I am problem driven and interested in a diverse set of phenomena originating in fluid mechanics and/or materials science. Typically, I consider analytical problems arising from the interaction of two (ore more) different materials, species, phases. In particular I work in the following fields:

Suspensions and their effective behaviour

I am generally interested in the rigorous derivation of mean-field  and hydrodynamic limits in the theory of flows containing many small particles. This includes phenomena like sedimentation, increase of viscosity, and active matter.  From a practical perspective it is prohibitive to resolve each individual particle (microscopic scale) and one is rather interested in describing the evolution on larger scales. Here, kinetic and hydrodynamic models, describing the evolution of the  phase space (mesoscopic scale) or spatial (macroscopic scale) density of the particles, come into play. My work regards the rigorous derivation of limits from the finer to coarser scales, thus justifying the use of kinetic and hydrodynamic models.

In [1,2] we have proven the first rigorous result on the local applicability of Einstein's formula for the effective viscosity of suspensions. The article [3] studies the influence of the effective viscosity on sedimentation. We prove existence of, and convergence to, an effective macroscopic model, the transport-Stokes equation. In [7] we prove that no mean-field equation for a Doi-like model of particle orientations can exist. It is necessary to consider higher order distribution functions. Most recently, in [8,10], we develop a rather complete theory for mean-field limits of particles with small inertia. We improve general mean-field theory for p-Wasserstein distances, and, using detailed understanding of the microscopic dynamics,  we show that the transport-Stokes equation is the macroscopic mean-field limit. In a second step we use this result to prove that the (mesoscopic) Vlasov-Stokes equation is a valid approximation and with smaller error than transport-Stokes. We also provide an improved hydrodynamic limit from Vlasov-Stokes to transport-Stokes.  

Collaborators in this area are: Barbara Niethammer, Richard Höfer, Amina Mecherbet

Models for phase separation and relaxation in non-convex energy landscapes

Models for phase separation play a major role in materials science and fluid dynamics.  Sharp-interface models consider pure phases, leading to geometric evolutions/dynamic free boundary problems while phase-field models allow for a local mixing of the phases, leading to 2nd and higher order nonlinear parabolic evolutionary PDE. In many cases these evolutions are driven by the reduction of surface area of the (diffuse) interface, and can be viewed as gradient flows.  In this setting I am interested in quantifying rates of convergence. This convergence may on the one hand happen for long times, where I consider relaxation rates to (meta-) stable configurations and metastable dynamics. On the other hand the convergence may happen from one model to another (for example phase-field to sharp-interface) through the vanishing of a small parameter.

In [4] we consider the Cahn-Hilliard equation with initial  data that have a finite L^1 distance to a so called bump - a (meta-)stable state with two transition layers. We prove optimal algebraic relaxation  towards the bump on the torus and on the line. This is another step towards a description that can handle coarsening events (the annihilation of two transition layers). The recent [9] establishes optimal relaxation rates for the Mullins-Sekerka evolution, a nonlocal free boundary problem. We prove generation of graph structure and optimal convergence to the planar interface starting from very rough initial data, that need neither need to be given by a graph nor need the initial phases be connected. We show that tools from geometric measure theory can be employed to obtain generation of graph structure from the dissipation of energy alone. 

Collaborators in this area are: Sarah Biesenbach, Felix Otto, Maria Westdickenberg

Variational approaches to Navier-Stokes and data-driven continuum mechanics

One reason for the mathematical problems associated to the Navier-Stokes equations is the lack of a Hamiltonian or dissipative structure for the equation. Typical issues are usually related to, and characterized by, the onset of turbulent behaviour. My goal is to understand these issues better by deriving variational characterizations for the (non-Newtonian) Navier-Stokes. At first sight unrelatedly, in the whole field of applied mathematics data-driven methods have become ubiquitous. On particular direction is to replace modelling assumptions, for example constitutive laws, by increasingly large data sets. Connecting both worlds, in [6], we have layed theoretical groundwork by establishing a variational framework for data-driven fluid mechanics in the stationary case in particular considering relaxation problems with semilinear constraint.

Collaborators in this area are: Christina Lienstromberg, Stefan Schiffer

Publications

Preprints

Peer-reviewed articles

Theses