Titles and Abstracts

Main Lectures

Juhi Jang

Vacuum Free Boundary Problems in Gas Dynamics

We will discuss recent progress on the vacuum free boundary problems arising in the dynamics of isolated gases with or without gravity. We will give an overview of the well-posedness and stability theory, and present some new results on waiting time solutions.

Chiara Saffirio

Uniqueness criteria for the Vlasov-Poisson system and applications to mean-field and semiclassical problems.

The Vlasov-Poisson system is a non-linear PDE describing the mean-field time-evolution of particles forming a plasma.

In this talk I will present uniqueness criteria for the Vlasov-Poisson equation, emerging as corollaries of stability estimates in strong or in weak topologies, and show how they serve as a guideline to solve mean-field and semiclassical problems. Different topologies will allow us to treat different classes of quantum states.

Li Wang

Learning-enhanced structure preserving particle methods for Landau equation

The Landau equation stands as one of the fundamental equations in kinetic theory and plays a key role in plasma physics. In this talk, I will present two particle approximations to the Landau equation, both of which leverage the gradient flow structure of the equation, with some assistance from deep learning techniques. While these approximations are primarily motivated by computational considerations, they may also offer new insights into the properties of the equation.

Jonathan Luk

The Einstein—massless Vlasov system and high-frequency gravitational waves

I will discuss a phenomenon where solutions to the Einstein—massless Vlasov system arise as limits of solutions to the Einstein vacuum equations. The phenomenon is closely related to a conjecture by Burnett, which suggested a spacetime metric is a high-frequency limit of solutions to the Einstein vacuum equations if and only if it is isometric to a solution to the Einstein—massless Vlasov system. Physically, this means that high-frequency gravitational waves do not interact with each other except through their effect on the geometry. In this talk, I will explain recent mathematical progress on Burnett's conjecture. In particular, I will present a recent result showing that a large class of small data U(1)-symmetric solutions to the Einstein—massless Vlasov system can be achieved as limits of solutions to the Einstein vacuum equations. Joint work with Cécile Huneau.

Yan Guo 

Ghost Effect and Perfect Conductor Boundary Condition 

We will present recent proof of validity of ghost effect in Boltzmann theory as well as stability of a collisional plasma confined within a perfect conductor.

Jacob Bedrossian

TBD

Zaher Hani

Long time justification of wave turbulence theory. 

We will discuss some recent joint works with Yu Deng (Chicago), in which we justify the

predictions of wave turbulence theory, including the derivation of the wave kinetic equation (WKE), on arbitrarily long time intervals covering the full lifespan of the kinetic equation. This result was the first of its kind, and paved the way for similar progress on the classical problem of deriving Boltzmann's equation on arbitrarily long times. 

Yu Deng

Hilbert's Sixth Problem: Derivation of the Boltzmann and fluid equations

We present recent works with Zaher Hani and Xiao Ma, in which we derive the Boltzmann equation from the hard sphere dynamics in the Boltzmann-Grad limit, for the full time range in which the (strong) solution to the Boltzmann equation exists. This is done in the Euclidean setting in any dimension $d\geq 2$, and in the periodic setting in dimensions $d\in\{2,3\}$. As a corollary, we also derive the corresponding fluid equations from the the hard sphere dynamics. This resolves Hilbert's Sixth Problem pertaining to the derivation of hydrodynamic equations from colliding particle systems, via the Boltzmann equation as the intermediate step.

Matthew Novack

Flexibility for an isotropic kinetic equation

We discuss flexibility phenomena (non-uniqueness, existence of non-trivial stationary solutions) in the context of an isotropic Landau-type kinetic equation. Based on ongoing joint work with Will Golding. 

Luis Silvestre

The monotonicity of the Fisher information for the Boltzmann equation

We discuss recent results showing that the Fisher information is monotone decreasing for the space homogeneous Boltzmann and Landau equations, and their consequences.

Short Talks

Patrick Flynn

Negative regularity mixing of passive scalars in stochastic fluid mechanics

Consider a passive scalar advected by a random vector field on a compact manifold, such as the solution to the 2D stochastic Navier-Stokes equation on a periodic box. In this talk, I discuss my work with J. Bedrossian and S. Punshon-Smith, where we show that if the passive scalar is initially in some negative regularity Sobolev space, then it will decay exponentially in the same space (in expectation). We prove this result using techniques from dynamical systems theory and semiclassical analysis. Going forward, we hope to apply this result to the problem of turbulence.

Sanchit Chaturvedi

Phase mixing in astrophysical plasmas with an external Kepler potential

In Newtonian gravity, a self-gravitating gas around a massive object such as a star or a planet is modeled via Vlasov Poisson equation with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in towards the object or escape to infinity. We focus on this regime and prove first a linear phase mixing result in 3D outside symmetry with exact Kepler potential. Then we also prove a long-time nonlinear phase mixing result in spherical symmetry. The mechanism is phenomenologically similar to Landau damping on a torus but mathematically the situation is quite a lot more complex. This is based on an upcoming joint work with Jonathan Luk at Stanford.

Timur Yastrzhembskiy

Asymptotic stability for the Vlasov-Maxwell-Landau system in a bounded domain

Following Prof. Yan Guo's presentation, I will highlight the key ideas in the proof of the asymptotic stability of Maxwellians for the Vlasov-Maxwell-Landau system in the presence of a perfectly conducting, specularly reflecting boundary. This is joint work with Hongjie Dong and Yan Guo.

Trinh Nguyen

Validity of Prandtl's boundary layer from the Boltzmann theory

We justify Prandtl equations and higher order Prandtl expansion from the hydrodynamic limit of the Boltzmann equations. Our fluid data is of the form shear flow, plus small order term in analytic spaces. The novelty lies in obtaining estimates for the linearized Boltzmann equation with a diffusive boundary condition around a Prandtl layer flow.

William Golding

Smoothing Effects in the Homogeneous Landau Equation

I will present several recent conditional regularity results for the homogeneous Landau equation, a fundamental kinetic model for collisional plasmas. These estimates have several implications for the existing theory including: expanding the existence framework for rough, slowly decaying initial data; extending uniqueness theorems to a larger class of solutions; and yielding sharper rates of convergence to equilibrium. I will also discuss the relevance of these regularity estimates to the full inhomogeneous model.

Dominic Wynter

Global Solutions for the Landau-Bose-Einstein Equation

Quantum kinetic theory, developing from the quantum Boltzmann equation, is a well-established field, with Fermi-Dirac statistics now fairly well understood. However, its mathematical treatment for Bose-Einstein statistics remains incomplete. For short-range interactions, the bosonic Boltzmann equation exhibits singularity formation, where solutions develop a Dirac mass at the origin, and stability and global well-posedness results are known near both singular and nonsingular equilibria. By contrast, long-range models like the bosonic Landau equation, introduced for self-gravitating bosons, remain largely unexplored mathematically. We construct global strong solutions near equilibria above the critical temperature and establish subexponential convergence to equilibrium, addressing challenges from the higher-order nonlinearity due to quantum effects and mass concentration near the critical temperature. This is a joint work with Maria Gualdani and Natasha Pavlovic.