Oberseminar Analysis Summer 2024

Recent years have seen tremendous progress regarding the derivation of effective models for sedimentation, in particular the transport-Stokes equation, from microscopic models. The strategy is usually to approximate the interaction of the microscopic particles through the fluid, which is very implicit and non-binary, by an explicit binary, albeit singular, interaction. In a second step one can use mean-field results for such explicit systems to conclude. However, this strategy requires good control of the minimal distance between the particles whence all known results require some unphysical condition on the scaling of the minimal distance of the particles. After reviewing some of the results, I will show how to overcome this limitation in the case of binary interacting systems with a non-attracting kernel and singularity just below the one of sedimentation. This is based on joint work with Richard Höfer (Regensburg).

Magnetohydrodynamics plays a crucial role in understanding the behavior of plasmas, electromagnetic fields, and fluid dynamics, providing a fundamental framework for studying phenomena in astrophysics, fusion energy, and space exploration. In this talk, we will present a short survey about the well-posedness of boundary value problems for steady ideal magnetic fluids. More precisely, we will first focus on the magneto-hydrostatic equations (in two and three dimensions) and conclude with some ongoing work related to the steady magneto-hydrodnamic equations.

Polchinski's equation is a partial differential equation (PDE) that describes the evolution of the effective action in studies of the renormalization group (RG) when a continuous scale decomposition is implemented. However, solving Polchinski's equation is a task as challenging as evaluating the functional integrals from which this equation was generated in the first place. The Majorant Method for fermions consists of a technique to control Polchinski's equation using another associated Hamilton-Jacobi equation and obtaining conditions for the existence and analyticity of the effective action. The method was introduced by D. Brydges and J. Wright in the late '80s, but its development contained a gap. In this talk, I will address the method and discuss how to overcome this gap. Some applications will also be discussed.

In this talk, I will present recent process on a joint work with Jani Lukkarinen.

We study the time-evolution of cumulants the kinetic energies in Kac's stochastic model for velocity exchange of N particles. Our aim is to quantify how fast – on a time scale in which the collision rate for each particle is of order one – these degrees of freedom become chaotic. Chaos here is understood in the sense of almost complete independence of the relevant dynamical random variables (kinetic energies of the particles) which we measure by the magnitude of their cumulants up to a finite, but arbitrary order.

Known spectral gap results imply that typical initial densities converge to uniform distribution on the constant energy sphere at (rescaled) times comparable to N, and the more recent entropy production arguments give similar time orders. We prove that the finite order cumulants converge to their small stationary values much faster, already at (rescaled) times of order (i) one or (ii) log(N), depending on the magnitude of the initial non-independence of the kinetic energies. The proof relies on stability analysis of the closed, but nonlinear, hierarchy of energy cumulants around the fixed point formed by their values in the stationary spherical distribution.

In phase-field models, like Cahn-Hilliard or Allen-Cahn equations with the adoption of singular potential, it is of paramount importance to assess that the solution stays uniformly away from the pure phases, instantaneously or at least eventually in time. This is useful to study higher-order regularization properties of the solution, as well as its longtime behavior. In this talk I would like to present some recent results, concerning the validity of this property in many phase-field models, from the Cahn-Hilliard equation (local and nonlocal) to the conserved Allen-Cahn equation. I will try to outline the most updated state of the art about the strict separation property, hopefully giving also some possible ideas towards the solution of some still open questions related to the topic.