RMU Seminar Solution Concepts in PDEs

I will discuss a new result on the existence of a family of compact subdomains of the flat cylinder for which the Neumann eigenvalue problem for the Laplacian admits eigenfunctions with constant Dirichlet values on the boundary. These domains have the property that their boundaries have nonconstant principal curvatures. In the context of ambient Riemannian manifolds, our construction provides the first examples of such domains whose boundaries are neither homogeneous nor isoparametric hypersurfaces. The underlying functional analytic approach we have developed overcomes an inherent loss of regularity of the problem in standard function spaces. With the help of this approach, we also construct a related family of subdomains of the 2-sphere. By this we disprove a conjecture of Souam from 2005.

This is joint work with M.M. Fall and I.A. Minlend.

This talk is concerned with directional differentiability results for solution operators of parabolic evolution variational inequalities. We discuss the (in large parts still open) problem of characterizing the directional derivatives of the solution map of a parabolic variational inequality by means of a suitably defined auxiliary problem. It is shown that the main difficulty that arises in this context is that the appearing directional derivatives are discontinuous as functions of time. This implies that standard weak formulations for evolutionary problems without any additional selection criteria are not able to provide a unique characterization. We present new results that show how this problem can be overcome for sweeping processes in the sense of Moreau by means of Kurzweil-Stieltjes integration theory and a vectorial version of Stampacchia’s lemma.

The talk concludes with a discussion of open questions and directions for future research.

In this talk, I will present a diagram-free approach to the theory of regularity structures. In particular, our method can be applied to show a priori estimates in Hölder spaces for renormalized, classically ill-defined quasilinear SPDEs in the subcritical regime. We first discuss a novel and efficient method to obtain (linear) Schauder estimates for germs which correspond to solutions of elliptic equations in anisotropic settings. The notion of a germ in regularity structures is a generalization of the standard Taylor polynomials. This method does not use kernel estimates, but is based on a scaling argument originally introduced by Simon in the classical case. I will then show how these linear estimates can be applied to derive estimates for the nonlinear problem via our diagram-free approach.

The talk is based on joint work with Scott Smith, and on joint work with Felix Otto, Scott Smith, and Hendrik Weber.

Partial differential equations (PDEs) are typically classified as elliptic, parabolic, or hyperbolic (with exceptions that need to be treated individually). While the close connections between elliptic and parabolic PDEs are well known, hyperbolic PDEs often stand apart and are approached with distinctly different techniques, both analytically and numerically. Recently, there has been growing interest in first-order hyperbolic approximations of PDEs, including classical elliptic equations like Poisson problems, parabolic problems such as the (fourth-order parabolic) Cahn-Hilliard equation, and dispersive water wave models like the Serre-Green-Naghdi equations. This talk offers a brief overview of some of these developments.

The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not 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 which lead to instabilities. It will turn out that these instabilities can lead to topology changes which can result in the splitting of so-called protocells.
In addition, we present numerical simulations which 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.

When a particle interacts with a quantized field, its effective mass increases: since the particle has to drag field deformations around with itself when it moves, it appears to be heavier than without the coupling. One effect of this phenomenon is that, when coupled to a quantum field, a quantum particle can have a bound state in a potential well that is too shallow to produce a bound state for the bare particle. This is referred to as enhanced binding.

The above reasoning is expected to be true on physical grounds, but enhanced binding has to be proved mathematically in concrete models. The task is to show that for a certain family of operators $H_\alpha$ (representing the quantum system, and parametrized by the coupling strength $\alpha$), $H_0$ does not have $L^2$ eigenvector, but $H_\alpha$ does for large enough $\alpha$.

Several approaches exist for this task. I will present one that relies probabilistic methods, more precisely on functional integrals and the Gaussian correllation inequality. I will first introduce the model, then motivate the use of functional integrals, and finally give an outline ofd the method.

This is based on joint work with Tobias Schmidt (Darmstadt) and Mark Sellke (Harvard).

Let us consider the Euler equations modeling the behavior of an incompressible, homogeneous, inviscid fluid. In the two-dimensional case, the Euler equations can be written in vorticity form as a continuity equation, in which the advecting velocity depends on the vorticity through an integral operator. In my talk, I will introduce several notions of weak solutions for the two-dimensional Euler equations in vorticity form: weak solutions, renormalized solutions, and vanishing-viscosity solutions. Relying on the linear theory for continuity equations with Sobolev velocity field by DiPerna and Lions, I will show that in the subcritical case weak solutions do not exhibit anomalies. In the supercritical case, I will show by means of a duality approach that the same holds for vanishing-viscosity solutions. This has some connections with the two-dimensional theory of turbulence of Kraichnan and Batchelor. If times allows, I will also comment on a stochastic approach which provides a convergence rate in the vanishing-viscosity limit.

The standard F-KPP (Fisher- Kolmogorov-Petrovsky-Piscounov) equation is a classical reaction-diffusion equation admitting so-called travelling wave solutions. In this talk, we will explain how certain interacting particle systems may be used to describe solutions of (inhomogeneous) F-KPP equations. We then use this description to obtain estimates on the transition front of such solutions.

Moreover, we will see how the Feynman-Kac formula (a tool from stochastic analysis) may be used to obtain estimates on solutions of F-KPP equations.

The talk is based on ideas from collaborations with A. Bovier, A. Drewitz and P. Hanigk.