Title/Abstract

Jose A. Carrillo

Mean-Field Derivation of the Space-Homogeneous Landau Equation via BBGKY Hierarchy Method

We consider the Kac particle system for the space-homogeneous Landau equation. For the Coulomb potential, we show that the Fisher information of the Liouville equation is monotonically decreasing in time. The monotonicity ensures the compactness to derive a weak solution of the Landau hierarchy. This is a work in collaboration with Shuchen Guo. For hard potentials, Shuchen Guo has shown that the propagation of exponential moments holds at the particle level. These moments bound ensures the uniqueness of weak solutions of the Landau hierarchy, which implies propagation of chaos for this range.

Hongjie Dong

Wellposedness and asymptotic stability for relativistic Vlasov-Maxwell-Landau system in general bounded domains

In this talk, I will present a joint work with Yan Guo and Timur Yastrzhembskiy in which we established a global well-posedness theory as well as asymptotic stability for the relativistic Vlasov-Maxwell-Landau system in a general 3D domain with a specularly reflective and perfectly conducting boundary.

Weinan E

Towards an Understanding of the Principles behind Deep Learning

The field of deep learning is evolving rapidly, driven by the availability of the vast amount of data and computing resources. Deep learning techniques have also evolved in several different ways, including different formulations such GAN and the diffusion model, different architecture such as CNN and transformers, and different training protocols such as BERT and GPT. This evolution has largely been empirical. Consequently there are a lot mysteries, surprises and “black magics” in this field. Is it possible to decipher some kind of guiding principles behind this? In this talk, we will discuss our thoughts along this line. Specifically, we will discuss how simple mathematical concepts such as complexity and stability can be used as guiding principles for designing and understanding neural network models.

David Gérard-Varet

Stability analysis of kinetic models for active particles

We shall review stability results for mesoscopic models of self-propulsing particles. These are nonlinear/non-local transport equations with the key feature that the usual velocity variable in R^3 is replaced by an orientation variable in the sphere S^2. This modifies substantially the mixing and enhanced dissipation phenomena and the subsequent mathematical analysis. The talk is based on joint works with Coti Zelati and Dietert and with Coti Zelati and Ertzbischoff 

Pierre Germain

Kinetic theory for oscillator chains

Oscillator chains are discrete nonlinear dispersive equations which can be thought of as one-dimensional models of crystals. The most famous instance is the Fermi-Pasta-Ulam-Tsingou model, but there is a great variety. I will give an overview of the associated kinetic theory and its hydrodynamic limits and emphasize the (few) parts which have been made rigorous. This is joint work with Joonhyun La and Angeliki Menegaki.

Seung Yeal Ha

Emergent behaviors of the inertial spin model and its extensions

In this talk, we discuss recent progress on the flocking estimates for the inertial spin(IS) model on the three-dimensional Euclidean space, and high-dimensional generalization of the three-dimensional one. For two-dimensional case, it reduces to the inertial Kuramoto model for synchronization. For quantitative flocking estimates, we derive a system of differential inequalities for relative kinetic and spin energies, and explicitly obtain exponential flocking under the proposed frameworks. We also introduce a high-dimensional inertial spin model which generalizes the inertial spin model on three-dimensional Euclidean space and study its emergent behaviors. For the generalization of the IS model to the high-dimensional Euclidean space, we replace the cross product by the multiplication via skew-symmetric matrix, and identify a new set of constants of motions which are conserved along the proposed model. We provide two frameworks leading to the collective behaviors of the high-dimensional IS model leading to qualitative and quantitative emergent dynamics in terms of system parameters and initial data. 

Zaher Hani

On Hilbert’s sixth problem: from particles to waves

In his sixth problem, Hilbert called for the derivation of the equations of fluid mechanics—such as the Euler and Navier-Stokes equations—from first principles, by rigorously justifying Boltzmann’s kinetic theory. This entails starting from Newton’s laws for a system of N particles and taking successive limits to first obtain Boltzmann’s kinetic equation, and then deriving the equations of fluid mechanics from it.

The major landmark in the early literature is the work of Oscar Lanford (1975), who provided the first rigorous derivation of the Boltzmann equation, albeit only for short times. Hilbert’s sixth problem, however, requires a long-time version of Lanford’s result, which remained open for decades.

In recent joint work with Yu Deng (University of Chicago) and Xiao Ma (University of Michigan), we extend Lanford’s theorem to long times—specifically, for as long as the solution of the Boltzmann equation exists. This allows us to fully carry out Hilbert’s program and derive the fluid equations in the Boltzmann–Grad limit.

The underlying strategy builds on earlier joint work with Yu Deng that resolved a parallel problem in which colliding particles are replaced by nonlinear waves, thereby establishing the mathematical foundations of wave turbulence theory. In this talk, we will review this progress and discuss some future directions.

Alexandru Ionescu

On the wave turbulence theory of 2D water waves 

I will talk about some recent work on the problem of establishing rigorously a wave turbulence theory for water waves systems. This is a classical problem in Mathematical Physics, going back to pioneering work of Hasselmann.

More precisely, I will discuss recent results on the long-time existence of solutions of water waves systems and on the well-posedness theory of the associated kinetic equations. To address the quasi-linear nature of the problems we propose a new mechanism, based on a combination of two main ingredients: (1) deterministic energy estimates for all solutions that are small in $L^\infty$-based norms, and (2) probabilistic arguments aimed at understanding propagation of randomness on long time intervals.

This is joint work with Yu Deng and Fabio Pusateri.

Shi Jin

Quantum Computation of Linear and Nonlinear Partial Differential Equations

Quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators. Nonlinearity and Nonunitarity are the main challenges for quantum simulations of PDEs.

For linear PDEs (and ODEs) Schroginerization provides a general method to unitarize them for quantum simulation. For nonlinear problems we first introduce our quantum algorithms for (nonlinear) Hamiton-Jacobi equations, for both multi-valued solutions and viscosity solutions that are needed beyond the formation of caustics. We also introduce quantum algorithms for Young measures associated with nonlinear PDEs, which are effective tools to compute weak solutions to nonlinear PDEs that have singular solutions such as shocks, caustics, physical instabilities or (random) uncertainties.

We also introduce “UnitaryLab”, which is an AI-powered research software package of quantum algorithms for scientific computing. 

Nader Masmoudi

TBA

Sung-Jin Oh

Accelerated Shock Formation for the Energy-Critical Euler-Poisson System

The gravitational Euler-Poisson system provides a basic model of a self-gravitating isolated star. For the energy-critical polytropic equation of state, there is an explicit radial steady-state solution called the Lane-Emden star, which features a fluid density that decays towards spatial infinity. I will present an ongoing work that describes the nonlinear phase space around this solution and proves the existence of a new instability mechanism: accelerated shock formation. This talk is based on an upcoming work in collaboration with Mahir Hadžić, Juhi Jang, and Ely Sandine.

Felix Otto

Capillarity-driven thin viscous films with thermal noise

We consider the evolution of the height of a liquid film on a planar solid substrate, driven by capillarity and limited by viscosity, yielding a height-dependent mobility. The fluctuation-dissipation principle guides how to introduce thermal forces.

Any spatial discretization of this singular stochastic PDE relies a choice of a numerical mobility, and leads to a large system of stochastic ODEs. The latter requires a non-obvious counter term that diverges as the grid size goes to zero, and which does not vanish when passing to the Stratonovich formulation.

The counter term can be derived from the informal gradient flow structure via the corresponding Fokker-Planck equation; it ensures a Riemannian covariance property and amounts to a Wick renormalization.

This is joint work with B.~Gess, R.~Gvalani, F.~Kunick, and -- in progress -- M.~Sauerbrey.

Jared Speck

The Global Structure of Stable, Unique, Maximal Globally Hyperbolic Developments for Shock-Forming 3D Compressible Fluids

I will discuss recent work, joint with Leo Abbrescia and Dongxiao Yu, on the 3D isentropic compressible Euler equations in spherical symmetry. Under any equation of state except for the Chaplygin gas, we study open sets of smooth “asymptotically flat” data on (\mathbb{R}^3) that are perturbations of a non-vacuum constant state. Our main result is that these data launch a unique Maximal Globally Hyperbolic Development (MGHD), where singularities develop on part of the boundary. These are the first stable, global-in-space blowup + uniqueness results of their type for any quasilinear hyperbolic PDE on (\mathbb{R}^3) with asymptotically flat data. Roughly speaking, an MGHD is a largest possible classical solution that is uniquely determined by the data. Explicit examples for related quasilinear hyperbolic PDEs show that MGHD-uniqueness cannot be determined locally in spacetime, and can in fact fail for some equations/some data. For these reasons, our proof of MGHD-uniqueness relies on the fact that we can construct it in its entirety and derive the complete structure of its boundary, all the way out to spatial infinity, and all the way down to the center of symmetry. In particular, the boundary of our MGHDs consists of smooth curves continuously joined at a corner: a singular boundary that extends out to spatial infinity, along which the fluid’s gradient blows up, and a Cauchy horizon that extends all the way to the center of symmetry, along which the solution remains smooth. The techniques we use are robust and are applicable to a wide class of quasilinear hyperbolic PDEs that fail to satisfy the null condition. 

Alexis Vasseur

Boundary vorticity estimate for Navier-Stokes and control of layer separation at the inviscid limit

We provide a new boundary estimate on the vorticity for the incompressible Navier-Stokes equation endowed with no-slip boundary condition. The estimate is scalable through the inviscid limit. It provides a control on the layer separation at the inviscid Kato double limit, which is consistent with the layer separation predictions via convex integration. 

Cédric Villani

TBA

Yanjin Wang

Stability of Shear Flows for 3D Euler

We will present our recent result of the stability of a large class of shear flows for the 3D Euler system with inflow boundary conditions in finite square or circular pipes. This is a joint work with Professor Yan Guo.

Yong Wang

Validity of Prandtl Expansion for Steady Compressible Navier-Stokes-Fourier Flows 

Studying steady flows is crucial in fluid dynamics for designing efficient systems (e.g., pipes, turbines, aircraft). Boundary layer study is fundamental in flying vessels surrounded by a compressible fluid, for which the classical boundary layer theory for incompressible fluid becomes inadequate. Despite its importance both from mathematical and physical standpoints, to our knowledge, there are very limited mathematical results to solve the steady compressible flows with both momentum and energy equations even for a finite Reynolds' number in the presence of mixed non-slip and in-flow boundary condition. 

Assume no-slip boundary conditions for the velocity field and either insulated or Dirichlet boundary conditions for the temperature field in a steady compressible fluid. In the inviscid limit $\v \rightarrow 0$ , we develop a mathematical framework for the uniform-in-$\v$ remainder estimate for the linear steady compressible Navier-Stokes-Fourier equations around a Prandtl layer profile with both velocity and thermal layers, which leads to the validity of the Prandtl layer expansion.

The talk is based on joint works with Professor Yan Guo.

Tong Yang

Critical function spaces with scaling invariance for kinetic equations

In this talk, we will discuss the well-posedness of the inhomogeneous Boltzmann and Landau equations in critical function spaces—a fundamental open problem in kinetic theory. We develop a new analytical framework to establish local well-posedness near a global Maxwellian for both equations, under the assumption that the initial perturbation is small in a critical norm. A major contribution lies in the introduction of a novel anisotropic norm adapted to the intrinsic scaling invariance of the equations, which provides precise control over the high-frequency behavior of solutions. By leveraging the regularizing effect and a decomposition of the linearized collision operator, we further extend the local solution globally in time and establish pointwise decay estimates. In addition, we will also present some work on the Green’s function in short time and the conditional regularity of the solutions. The talk is based on a joint work with Ke Chen and Quoc-Hung Nguyen.