Titles and Abstracts
(Listed in order of programme)
Cavitation and Concentration in the Solutions of the Compressible Euler Equations and Related Nonlinear Equations
Ryan Unger (Princeton/Cambridge)
Retiring the third law of black hole thermodynamics
Dr Yuzhe Zhu (Cambridge)
Hypocoercive and hypoelliptic properties for kinetic equations in bounded domains
Amélie Loher (Cambridge)
Global existence for Landau with small initial data
Dr Benjamin Fehrman (Oxford)
Non-equilibrium fluctuations and parabolic-hyperbolic PDE with irregular drift
Andrea Clini (Oxford)
Fluctuations of interacting particle systems and conservative stochastic PDE
Dr Immanuel Ben Porat (Oxford)
The graph limit: time dependent weights, singular interactions and link with the mean field limit
Alex Cliffe (Oxford)
Two-Dimensional Riemann Problems in Gas Dynamics
Prof Claude Warnick (Cambridge)
Effective Field Theories
Dr Tobias Barker (Bath)
On symmetry breaking for the Navier-Stokes equations
Abstracts
Cavitation and Concentration in the Solutions of the Compressible Euler Equations and Related Nonlinear Equations
In this talk, we will discuss the intrinsic phenomena of cavitation/decavitation and concentration/deconcentration in the entropy solutions of the compressible Euler equations, the compressible Euler-Poisson equations, and related nonlinear PDEs, which are fundamental to understanding the well-posedness and solution behaviour for nonlinear PDEs. We will start to discuss the formation process of cavitation and concentration in the entropy solutions of the isentropic Euler equations with respect to the initial data and the vanishing pressure limit. Then we will analyze a longstanding fundamental problem in fluid dynamics: Does the concentration occur generically so that the density develops into a Dirac measure at the origin in spherically symmetric entropy solutions of the multi-dimensional compressible Euler equations and related nonlinear PDEs? We will report our recent results and approaches developed for solving this longstanding open problem for the Euler equations, the Euler-Poisson equations, and related nonlinear PDEs, and discuss its close connections with entropy methods and the theory of divergence-measure fields. Further related topics, perspectives, and open problems will also be addressed.
Ryan Unger (Princeton/Cambridge)
Retiring the third law of black hole thermodynamics
In this short talk I will present a disproof of the celebrated ``third law of black hole thermodynamics.'' This is joint work with Christoph Kehle.
Hypocoercive and hypoelliptic properties for kinetic equations in bounded domains
We consider dissipative kinetic equations in bounded spatial domains with general boundary conditions that may not converse mass. We first discuss its hypocoercive property concerning the long-time behaviour of solutions. We then study the regularization property of solutions to hypoelliptic kinetic equations to upgrade the hypocoercive result.
Global existence for Landau with small initial data
We consider the Landau equation with Coulomb potential in the spatially homogeneous case. We show that for initial data near equilibrium in $L^p$ with $p>3/2$, classical solutions exist for all times and are unique.
Non-equilibrium fluctuations and parabolic-hyperbolic PDE with irregular drift
Non-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 far-from-equilibrium. In this talk, we will focus particularly on the fluctuations of the zero range process about its hydrodynamic limit. 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.
Fluctuations of interacting particle systmes and conservative stochastic PDE
Interacting particle systems have found diverse applications in mathematics and several related fields, including statistical physics, population dynamics, and machine learning. The large-scale behavior of these systems is essentially deterministic and is characterized by the solution to a nonlinear diffusion equation. This PDE furnishes a correct description of the particle system up to order zero. However, the particle process does exhibit large fluctuations away from its mean, i.e. from the solution of this PDE. Such deviations, though rare, can have significant consequences---such as a concentration of energy or the appearance of a vacuum---which make them important to understand and simulate. In particular, their behavior can be analyzed in terms of large deviation principles and central limit theorems, which respectively provide a sharp understanding of the occurrence of such fluctuations and a description of the particle system which is now correct up to order one. Both these approaches involve deterministic and stochastic variants of the above PDE.
In this talk, we first discuss these probabilistic concepts in some simpler settings and then we introduce a continuum model to simulate rare events of particle systems. Namely, we consider a precise stochastic version of the aforementioned limiting PDE, whose well-posedness has been recently obtained, and we show that the small-noise fluctuations of this SPDE around the deterministic zero-noise limit --- i.e. around the initial PDE --- behave as the fluctuations of the particle system: that is, they satisfy an identical large deviation principle and an identical central limit theorem. The talk is based on joint work with Benjamin Fehrman and on the fundamental previous work of B. Fehrman, B. Gess and N. Dirr.
Dr Immanuel Ben Porat (Oxford)
The graph limit: time dependent weights, singular interactions and link with the mean field limit
A gentle invitation to the graph limit regime, which is a family relative of the more well studied mean field limit regime. The talk would be mostly based on the approach taken by N. Ayi and N. Pouradier Duteil, which considers the case where the weights evolve in time according to some ODE.
The question of how singular potentials may effect the limit might also be addressed, and is related to joint work in progress with J. A. Carrillo.
Two-Dimensional Riemann Problems in Gas Dynamics
The Riemann problem is the most simple initial value problem having discontinuous initial data which are piecewise constant and invariant under scaling. For systems of conservation laws in one space dimension, the solutions of Riemann problems form the building blocks for the general initial value problem, and the theory is well developed thanks to fundamental contributions from Lax (1957) and Glimm (1965). However, in two space dimensions, rigorous analytic results are few and far between, and many basic questions remain unanswered. We discuss recent progress in this area, focusing mainly on the two-dimensional shock reflection/diffraction problems, which can be reformulated as free boundary problems for a nonlinear PDE of mixed hyperbolic-elliptic type. Moreover, we present our latest result: the existence of a global four-shock interaction configuration in potential flow. Further details may be found in Chen-Feldman (2018) and Chen-AC-Huang-Liu-Wang (preprint: 2023).
Prof Claude Warnick (Cambridge)
Effective Field Theories
Often in a physical problem there are some degrees of freedom that we treat as unobservable because the energy required to excite them is much greater than the energy scale of the problem. For example, when modelling a pendulum with a rigid arm we ignore the vibrational modes of the arm. Or in quantum physics it may be that the energy scales available to us, for example in a collider, are not sufficient to create a particular heavy particle. These unobservable degrees of freedom can still affect the physics that we do observe at subleading order. A tool that is often used by physicists to study this is the machinery of effective field theory (EFT). In this talk I will discuss work with Harvey Reall in which we studied an EFT for a model problem and gave a prescription to handle the ill-posed PDE problems that typically arise.
On symmetry breaking for the Navier-Stokes equations
Motivated by an open question posed by Chemin, Zhang and Zhang, we investigate quantitative symmetry breaking for the 3D incompressible Navier-Stokes equations with initial third component of the velocity equal to zero. Specifically, I will discuss: (i) isotropic norm inflation with unfavourable initial pressure gradient, (ii) quantitative symmetry breaking with favourable initial pressure gradient. Joint work with Christophe Prange (CNRS, Cergy) and Jin Tan (Cergy).