Analysis of Partial Differential Equations

Partial differential equations (PDEs) lie at the heart of many different fields of Mathematics and Physics: Complex Analysis, Minimal Surfaces, Kähler and Einstein Geometry, Geometric Flows, Hydrodynamics, Elasticity, General Relativity, Electrodynamics and Nonrelativistic Quantum Mechanics, to name only a few. Given the enormous variety of phenomena that they model, it is perhaps not surprising that no general theory concerning solvability and dynamics of their solutions is known. While there is a common set of techniques that analysts use (which naturally invokes methods from Functional Analysis, Geometry, Harmonic Analysis and Probability), one typically needs to unravel the specific structure of the PDE at hand to tackle questions like existence and uniqueness of solutions, stability and rigidity phenomena, formation of singularities and patterns etc. This may involve (but is not limited to) developing a priori estimates, adapted function spaces, appropriate theories of weak solutions, microlocal analysis, variational techniques and much more.

In Münster, (non-linear) PDEs are studied in a wide variety of contexts, which are naturally intertwined:

  • PDEs arising in general relativity

    The Einstein equations of general relativity are hyperbolic evolution equations. Here we are interested in understanding the long term dynamics of the equations near black hole solutions (stability), as well as the formation of singularities and the rigidity of solutions under symmetry assumptions. Symmetry reductions of the Einstein equations often lead to interesting other PDEs (e.g. elliptic or harmonic map type) that appear independently in other fields. Finally, as the Einstein equations themselves are hyperbolic, one is often lead to understand the long term behaviour of ((non)-linear) wave equations in curved geometries.

  • PDEs arising in fluid dynamics

    The Euler and Navier-Stokes equations are the cornerstone of fluid dynamics. Our research focuses on the evolution and stability of vortex structures, such as laments and sheets, which arise in real-world phenomena like tornadoes and bubble rings. A key objective is to reduce the complex dynamics of fluid motion to the evolution of simpler geometric objects, such as curves. Additionally, we study mixing processes driven by turbulent flowsor vortex dynamics, often incorporating stochastic  elements into simplified models in order to capture key features of turbulence.

  • PDEs arising in differential geometry

    Here a central theme is understanding the long term dynamics and the formation of singularities for the (parabolic) Ricci-flow. Common techniques involve the maximum principle, energy estimates and the exploitation of monotone quantities (or geometric inequalities that are preserved) along the flow. Another equation that we study is the Monge-Ampere equation appearing in Kähler geometry for which we are, for instance, interested in the regularity theory of solutions. See Differential Geometry for more details.

  • PDEs arising in mathematical modelling

    The theoretical analysis of PDE-models is central also in mathematical modelling. Many real world phenomena are of multiscale nature and a small number of fundamental effects is responsible for a large number of observables. The theoretical analysis of related PDE-models allows for the derivation of rigorous predictions and provides the connection with a few fundamental model parameters.

    Mathematically we are interested e.g. in kinetic equations, their approximations by parabolic equations and BGK approximations, cross-diffusion systems, and the existence, uniqueness, and longtime behavior of solutions. Asymptotics which maintain main properties of the original model, invariants of model equations, singularity formation and pattern formation are analyzed. From the applied side we are interested e.g. in gas mixtures, bio-chemical reactions, semiconductor devices, attraction-repulsion effects, cell motility, population dynamics, and epidemiology.