Noise plays a fundamental role in fluid dynamics: already in the 19th century, J. Boussinesq conjectured that the turbulent character of fluids should be captured by a stochastic description. In recent years, a particular noise of transport type has received substantial attention in the literature and in this talk we will review a selection of results: on the one hand, a scaling limit under which the noise delays potential blow-up by enhancing diffusion of the system; on the other hand, the method of convex integration giving rise to non-uniqueness of stochastic solutions. Finally we will touch upon corresponding future research directions.
We prove a central limit error bound for convolution powers of laws with finite moments of order $r \in \mathopen]2, 3\mathclose]$, taking a closeness of the laws to normality into account. Up to a universal constant, this generalises the case of $r=3$ of the sharpening of the Berry (1941) - Esseen (1663) theorem obtained by Mattner (2024), namely by sharpening here the Katz (1963) error bound for the i.i.d. case of Lyapunov's (1901) theorem. For this purpose, we introduce Zolotarev's (1976) $\zeta$ distances for probability measures and a certain variant thereof due to Senatov (1980). Our proof uses a convolution inequality obtained by Mattner (2024) and a partial generalisation of a theorem of Senatov and Zolotarev (1986).
We consider an incompressible Stokes fluid contained in a box $B$ that flows around an obstacle $K\subset B$ with a Navier boundary condition on $\partial K$. I will present existence and partial regularity results for the minimization of the drag of $K$ among all obstacles of given volume.
I will discuss the scaling limits of a class of deterministic surface growth models on lattices, which includes “Edwards-Wilkinson growth” (the height of the surface evolves according to the discrete heat equation) and “Solid-on-Solid growth” (the height of the surface is the value function of a two-player, tug-of-war game). The novelty from a PDE point-of-view is, with the exception of the Edwards-Wilkinson case, the limiting evolution is determined by a fully nonlinear parabolic PDE with discontinuous gradient dependence. I will try to explain how the discontinuities arise as a result of the geometry of the underlying lattice. This insight can then be exploited to develop a theory of viscosity solutions for such strange PDE, which I will also briefly describe. Finally, I will conclude by describing some other applications, future directions, and open problems.
In this talk, I will discuss the birth-death dynamics for sampling multimodal probability distributions, which is the spherical Hellinger gradient flow of relative entropy. The advantage of the birth-death dynamics is that, unlike any local dynamics such as Langevin dynamics, it allows global movement of mass directly from one mode to another, without the difficulty of going through low probability regions. We prove that the birth death dynamics converges to the unique invariant measure with a uniform rate under some mild conditions, showing its potential of overcoming metastability. We will also show that on torus, the kernelized dynamics, which is used for numerical simulation, Gamma-converges to the idealized dynamics as the kernel bandwidth shrinks to zero. Joint work with Yulong Lu (UMass Amherst) and Dejan Slepcev (CMU).
In this talk, I will discuss partial regularity theory for the Navier-Stokes equations in dimensions 3 and higher. I will also discuss the limitations of classical regularity theory when dealing with supercritical PDE systems of parabolic type, taking the Navier-Stokes equations as an example.
Transport properties of Gibbs measures under several dynamical systems have been studied in probability theory. In this talk, we study the Gibbs measure with log-correlated base Gaussian fields on functions/distributions. We first discuss the (non-)construction of the Gibbs measure and then study invariance, and quasi-invariance of the measures along the flow of dispersive Hamiltonian PDEs (like Schr¨odinger and wave equations).
You want to know what our new postdocs and group leaders are working on? What expertise they do have? Find out if you have some joint interests and visit our "New Faces Seminar", All (5) new postdocs staying at the institute for more than 18 months will give a short presentation of what they are doing and what their scientific interests are.
