# Workshop on partial differential equations related to calculus of variations, probability, and fluid dynamics

## Abstracts for the talks

**Nikolay Barashkov ***University of Helsinki***Segal Gluing for Φ _{3}^{4}**

Segal gluing is a property of Quantum Field theories relating partition functions of the theory under gluing of Euclidean Spacetimes. In this talk I will describe how to define and establish this property for the Phi^4_3 measure on cylinders. This is joint work with T. Gunaratnam.

**Angeliki Menegaki ***IHES, Université Paris-Saclay***Spectral gap for harmonic and weakly anharmonic chain of oscillators**

We consider one-dimensional chains and multi-dimensional networks of harmonic oscillators coupled to two Langevin heat reservoirs at different temperatures. Each particle interacts with its nearest neighbours by harmonic potentials and all individual particles are confined by harmonic potentials, too. In previous works we investigated the sharp N-particle dependence of the spectral gap of the associated generator in different physical scenarios and for different spatial dimensions. We also obtained estimates on the gap after perturbing weakly the quadratic potentials, through a Log-Sobolev Inequality. In this talk I will present new results on the behaviour of the spectral gap when considering longer-range interactions in the purely harmonic chain. In particular, depending on the strength of the longer-range interaction, there are different regimes appearing where the gap drastically changes behaviour but even the hypoellipticity of the operator breaks down. Parts of this talk are joint works with Simon Becker (ETH).

**Evan Miller ***University of British Columbia, and Pacific Institute for the Mathematical Sciences***On the regularity of axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions**

In this talk, I will discuss the axisymmetric, swirl-free Euler equation in four and higher dimensions. I will show that in four and higher dimensions the axisymmetric, swirl-free Euler equation has properties which could allow finite-time singularity formation of a form that is excluded in three dimensions. I will also discuss a model equation that is obtained by taking the infinite-dimensional limit of the vorticity equation in this setup. This model exhibits finite-time blowup of a Burgers shock type. The blowup result for the infinite dimensional model equation strongly suggests a mechanism for the finite-time blowup of smooth solutions of the Euler equation in sufficiently high dimensions.

**Chiara Rigoni ***University of Vienna***Tamed spaces - Dirichlet spaces with distribution-valued Ricci bounds**

In this talk we develop the theory of tamed spaces, which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature, and investigate these from an Eulerian point of view. To this end we introduce singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-Émery curvature-dimension condition.

This is a joint work with M. Erbar, K.-T. Sturm and L. Tamanini.

**Tommaso Rosati ***University of Warwick***Lyapunov exponents for SPDEs beyond order preservation**

We review some results regarding the study of Lyapunov exponents of stochastic partial differential equations (SPDEs) on finite volume and present a new approach through a dynamic separation of scales to treat SPDEs beyond order preservation. Joint work with Martin Hairer.

**Florian Schweiger ***Weizmann Institute of Science***Gaussian random interface models and discrete Green’s functions**

Random interface models describe microscopic fluctuations of the boundary layer between two different substances. Mathematically, they are given by a probability measure on a suitable space of height functions. In several important cases, this measure is Gaussian, and its covariance can be understood as the Green’s function of some elliptic differential operator (or of its discretization).

In the talk, I will explain this connection and discuss a few examples. I will then describe how one can obtain estimates for these Green’s functions using tools from PDE theory and numerical analysis, and how these estimates can be used to establish some properties of the corresponding interface model.

## Date and Location

**February 01, 2023**

Max Planck Institute for Mathematics in the Sciences

E1 05 (Leibniz-Saal)

Inselstr. 22

04103 Leipzig

## Scientific Organizers

**Felix Otto**

MPI for Mathematics in the Sciences**László Székelyhidi**

MPI for Mathematics in the Sciences

## Administrative Contact

**Anne Dornfeld**

Contact by Email

**Katja Heid**

Contact by Email