Schedule for: 24w5192 - Partial Differential Equations: Deterministic and Probabilistic

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

A brief introduction with important logistical information, technology instruction, and opportunity for participants to ask questions.

When you hit a drum, the sound it makes is a mix of overtones with frequencies corresponding to the drum's Laplace eigenvalues. A classic paper by Kac [“Can one hear the shape of a drum?” 1966] asks if the frequencies of these overtones uniquely determine the shape of the drum head. This question is still richly studied. We recently posed a related question: Can one hear where a drum is struck? Imagine you know a drum's shape. Could you determine where it is struck, up to symmetry, by listening also to the amplitudes of these overtones? In this talk, I will state this problem precisely, give additional physical interpretations, work some examples, and share our results so far.

Inverse random scattering problems with a random source or potential will be introduced for time-harmonic stochastic wave equations. The unknown random source or potential is assumed to be a generalized fractional Gaussian random field. With information of the data observed in a bounded domain, the strength of the random source or potential is shown to be uniquely determined by a single realization of the magnitude of the wave field averaged over the frequency band almost surely.

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

The aim of these lectures is to introduce various aspects of stochastic quantisation, including parabolic stochastic quantisation, variational approaches, etc.

In this series of lectures we will survey recent developments on the study of random data problems for nonlinear (dispersive) PDEs. Topics include probabilistic well-posedness, invariant measures, and off-equilibrium statistics (wave turbulence).

I will discuss a model of interacting particles in continuous space which is reversible with respect to Poisson point measures with constant density. Similar discrete models are known to "homogenize", in the sense that the evolution density profile can be approximated by the solution to a partial differential equation over large scales. Using ideas from the corresponding problem for elliptic equations, I will present some results that make this approximation quantitative. Based on joint works with Arianna Giunti, Chenlin Gu and Maximilian Nitzschner.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

I will present a finite time blow-up result of the Calogero–Moser derivative nonlinear Schrödinger equation (CM-DNLS). It is an L^2-critical nonlinear Schrödinger equation with explicit solitons, self-duality, and pseudo-conformal symmetry. More importantly, this equation is known to be completely integrable in the Hardy space and the solutions in this class are referred to as chiral solutions. A rigorous PDE analysis of this equation with complete integrability was recently initiated by Gérard and Lenzmann. Our main result constructs smooth, chiral, and finite energy finite-time blow-up solutions with mass arbitrarily close to that of soliton, adressing the global regularity question for chiral solutions raised by Gérard and Lenzmann. Our proof also gives a construction of a codimension one set of smooth finite energy initial data (but without addressing chirality) leading to the same blow-up dynamics. Our blow-up construction might also be contrasted with the global well-posedness of the derivative nonlinear Schrödinger equation (DNLS), which is another integrable L^2-critical Schrödinger equation. This talk is based on a joint work with Kihyun Kim and Taegyu Kim, arXiv:2404.09603.

This talk presents a quantitative homogenization for non-gradient exclusion process. Compared to the previous work by Giunti-Gu-Mourrat' 22, the new challenges here come from the hard core constraint of the particle number and the curse of dimension, and I will explain how to overcome them by a new coarse-grained strategy. As an application, our result can be integrated into the classical work Funaki-Uchiyama-Yau' 96 and yield a quantitative hydrodynamic limit. This talk is based on a joint work with Tadahisa Funaki (BIMSA) and Han Wang (Qiuzhen College, Tsinghua University).

We consider the recovery of coefficients in hyperbolic equations from boundary measurements. In this lecture, we explain the idea of geometric optics solution. We discuss recent advances for the related geometric inverse problems, and report new developments for solving the inverse problem with nonlinearities.

We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics, i.e., asymptotic compactness in dynamical systems, global stability of evolution equations, and localized control problems. As an initial application, we exploit the exponential mixing of random nonlinear wave equations with degenerate damping, critical nonlinearity, and physically localized noise. This is a joint work with Ziyu Liu, Dongyi Wei, Zhifei Zhang, and Jia-Cheng Zhao.

In this talk, I will discuss my studies on geometric singularity models (geometric type Liouville theorems) and select topics from mean curvature flows, and Allen-Cahn equations with minimal surfaces. More specifically, I will talk about classification of ancient solutions of mean curvature flow, and survey recent progress on the regularity and singularity structure of limiting minimal interfaces generated by Allen Cahn equations on manifolds with boundary if time permitted.

The aim of these lectures is to introduce various aspects of stochastic quantisation, including parabolic stochastic quantisation, variational approaches, etc.

Far-from-equilibrium behavior in physical systems is widespread. A statistical description of these events is provided by macroscopic fluctuation theory, a framework for non-equilibrium statistical mechanics that postulates a formula for the probability of a space-time fluctuation based on the constitutive equations of the system. This formula is formally obtained via a zero noise large deviations principle for the associated fluctuating hydrodynamics, which postulates a conservative, singular stochastic PDE to describe the system out-of equilibrium. In this talk, we will focus particularly on the fluctuations of certain interacting particle processes about their hydrodynamic limits. We will show how the associated MFT and fluctuating hydrodynamics lead to a class of conservative SPDEs with irregular coefficients, and how the study of large deviations principles for the particles processes and SPDEs leads to the analysis of parabolic-hyperbolic PDEs in energy critical spaces. The analysis makes rigorous the connection between MFT and fluctuating hydrodynamics in this setting, and provides a positive answer to a long-standing open problem for the large deviations of the zero range process.

Cosmic Microwave Background (CMB) is the thermal radiation remnant from the Big Bang. It is a primary source of information regarding the early universe. An outstanding question is what information can be inferred from it. In this talk, we use the kinetic model and show some inverse results. Mathematically, we study an inverse source problem for the linear Boltzmann equation and prove that the source term which is connected to the metric perturbation can be stably determined from the photon observation for generic absorption coefficients and scattering kernels. Our proof relies on the microlocal analysis of an integral transform in Lorentzian geometry (the light ray transform), and its interplay with wave equations.

In this series of lectures we will survey recent developments on the study of random data problems for nonlinear (dispersive) PDEs. Topics include probabilistic well-posedness, invariant measures, and off-equilibrium statistics (wave turbulence).

We first discuss the separation of particles into sparse and dense phases in Glauber-Kawasaki dynamics of non-gradient type and show that the phase-separating interface evolves according to the anisotropic curvature flow. The result explained by Chenlin Gu in the second lecture plays a crucial role to give a bound for Glauber rate. Then, we study the fluctuation of the interface in a simple situation and derive a linear SPDE via the Boltzmann-Gibbs principle. We also discuss heuristic derivation of nonlinear SPDEs.

In this series of lectures we will survey recent developments on the study of random data problems for nonlinear (dispersive) PDEs. Topics include probabilistic well-posedness, invariant measures, and off-equilibrium statistics (wave turbulence).

We consider the dynamics of a Bose–Einstein condensate in a repulsive interacting Bose gas and prove rigorously that the dynamics of the system, in the hard sphere scaling regimes where the number of particles tends to infinity and the Planck’s constant becomes negligible, are governed by the compressible Euler equations with internal pressure, proportional to the scattering length, coming from the microscopic interaction of the particles. We establish strong and quantitative microscopic-to-macroscopic convergence of mass and momentum densities up to the first blow-up time of the compressible Euler equations.

Calderón's inverse problem asks whether one can determine the conductivity of a medium by making voltage and current measurements at the boundary. This question arises in several areas of applications including medical imaging and geophysics. I will report on some of the progress that has been made on this problem since Calderon proposed it, including recent developments on similar problems for nonlinear equations and nonlocal operators. We will also discuss several open problems.

The aim of these lectures is to introduce various aspects of stochastic quantisation, including parabolic stochastic quantisation, variational approaches, etc.

Nonlinear phenomena of waves appear in many applications, for example, ultrasound imaging, optics and elasticity. I will discuss several inverse boundary value problems for wave equations. The main tool for the study is the analysis of nonlinear interactions of distorted plane waves or Gaussian beams.

Calderón's inverse problem asks whether one can determine the conductivity of a medium by making voltage and current measurements at the boundary. This question arises in several areas of applications including medical imaging and geophysics. I will report on some of the progress that has been made on this problem since Calderon proposed it, including recent developments on similar problems for nonlinear equations and nonlocal operators. We will also discuss several open problems.