Schedule for: 23w5002 - New Trends in Fluids and Collective Dynamics

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

PDC Lounge

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

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

We consider Bayesian data assimilation for time-evolution PDEs, for which the underlying forward problem may be very unstable or ill-posed. We formulate assumptions on the forward solution operator of such PDEs under which stability of the posterior measure with respect to perturbations of the noisy measurements can be proved. We also provide quantitative estimates on the convergence of approximate Bayesian filtering distributions computed from numerical approximations. For the Navier-Stokes equations, our results imply uniform stability of the filtering problem even at arbitrarily small viscosity, when the underlying forward problem may become ill-posed, as well as the compactness of numerical approximations in a suitable metric on time-parametrized probability measures. This is a joint work with Samuel Lanthaler and Siddhartha Mishra.

A lubrication model can be used to describe the dynamics of a weakly volatile viscous fluid layer on a hydrophobic substrate. Thin layers of the fluid are unstable to perturbations and break up into slowly evolving interacting droplets. In this talk, we will present a reduced-order dynamical system derived from the lubrication model based on the nearest-neighbour droplet interactions in the weak condensation limit. Dynamics for periodic arrays of identical drops and pairwise droplet interactions are investigated which provide insights to the coarsening dynamics of a large droplet system. Weak condensation is shown to be a singular perturbation, fundamentally changing the long-time coarsening dynamics for the droplets and the overall mass of the fluid in two additional regimes of long-time dynamics. This is joint work with Thomas Witelski.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet at Vistas for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neurons and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou

Given a desired target distribution and an initial guess of its samples, what is the best way to evolve the locations of the samples so that they accurately represent the desired distribution? A classical solution to this problem is to evolve the samples according to Langevin dynamics, a stochastic particle method for the Fokker-Planck equation. In today’s talk, I will contrast this classical approach with two novel deterministic approaches based on nonlocal particle methods: (1) a nonlocal approximation of dynamic optimal transport, with state and control constraints, and (2) a nonlocal approximation of degenerate diffusion equations. I will present recent work analyzing the convergence properties of each method, alongside numerical examples illustrating their behavior in practice. This is based on joint works with Karthik Elamvazhuthi, Matt Haberland, Matt Jacobs, Harlin Lee,and Olga Turanova.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we consider the 2D incompressible Boussinesq equation without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global-in-time smooth solutions where the H^1 norm of density grows to infinity algebraically in time. For the inviscid equation in the strip, we construct examples whose vorticity grows at least like t^3 and gradient of density grows at least like t^2 during the existence of a smooth solution. These growth results work for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth. This is a joint work with Alexander Kiselev and Jaemin Park.

In this talk we will consider the existence of time periodic solutions arbitrarily close to shear flows for the 2D incompressible Euler equations. In particular, we will present some results concerning the existence of such solutions near Couette, Taylor-Couette and Poiseuille flows. In the first part of the talk, we will introduce the problem and review some well-known results on this subject. In the second we will outline some of the ideas underlying the construction of our time periodic solutions. Joint work with Angel Castro.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will provide an overview of recent advancements in the field of local and global well-posedness theories concerning the compressible Euler system with singular velocity alignment. This system, commonly referred to as the Euler-alignment system, serves as a mathematical model for studying the collective behavior of animal flocks. My discussion will primarily focus on two specific scenarios: (1) the establishment of global well-posedness for small rough initial data that reside within critical Besov spaces, and (2) the achievement of global well-posedness for large smooth initial data in suitable multi-dimensional settings.

In this talk, we will see that certain singular coherent structures in 2D Euler equation propagates. Also, we will see that these solutions can be approximated by solutions of Boltzmann equation. The talk is based on a joint work with Theodore Drivas and Tarek Elgindi, and a joint work with Chanwoo Kim.

Motivated by motion of myxobacteria, we review several kinetic approaches for modeling motion of self-propelled, interacting rods. We will focus on collisional models of Boltzmann type and discuss the derivation of the governing equations, the range of their validity, and present some analytical and numerical results. We will show that collisional models have natural connection to classical mean-field models of nematic alignment.

We will go over the main ideas used in showing that as long as a super-critical Holder semi norm of the classical solution to the incompressible Navier-Stokes system is under control, the solution remains smooth. The key idea is exploiting an enhanced regularity effect coming from the transport term at the level of a simple one-dimensional drift-diffusion equation, allowing us to break the criticality barrier. We then employ ideas introduced by Kiselev, Nazarov, Volberg, and Shterenberg to propagate this to abstract drift-diffusion equations, providing to our knowledge the very first reasonable extension of the celebrated parabolic regularity result of Nash to an equation that is not in divergence form. Such an approach coupled with subtle pressure estimates due to Silvestre also allows us to rigorously treat the incompressible Navier-Stokes as a perturbation of drift-diffusion, obtaining a super-critical regularity criterion.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Chemotactic blow up in the context of the Patlak-Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of fluid advection can arrest singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong - an effect that is conjectured to hold for more general classes of nonlinear PDE. In this talk, I will discuss Patlak-Keller-Segel equation coupled with fluid flow obeying Darcy's law via buoyancy force. It turns out that in this case, the singularity formation is suppressed by arbitrarily weak coupling. The talk is based on joint work with Zhongtian Hu and Yao Yao.

The Euler Alignment system is a hydrodynamic PDE version of the celebrated Cucker-Smale ODE's of collective behavior. Together with Changhui Tan (University of South Carolina), we developed a theory of weak solutions in 1D, which provide a uniquely determined way to evolve the dynamics after a blowup. Inspired by Brenier and Grenier's work on the pressureless Euler equations, we show that the dynamics of interest are captured by a nonlocal scalar balance law, the unique entropy solution of which we generate through a discretization involving the "sticky particle Cucker-Smale" system. In this talk, we will discuss the formation of clusters of mass in the Euler Alignment system, and we will describe how to predict these clusters using the flux from the associated scalar balance law.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Machine learning and its applications in AI have entered into a new stage in their development: while the use of AI algorithms is widespread and will continue expanding, it is imperative to ask how can we guarantee that as these algorithms penetrate into more domains of our lives they will also be sensitive to privacy concerns, make fair decisions, and be both reliable and robust to data corruption. Are we ready to certify when a given algorithm complies with specific requirements and behaves in the way it is intended to? In this talk, I will discuss adversarial machine learning in supervised learning and clustered federated learning, two examples of ML settings where model accuracy is not the sole criterion for training learning systems. I will present novel approaches for the training of models in these two settings that rely on the use of particle dynamics and their analysis. Our solution to the first problem is inspired by the literature of gradient flows in the space of probability measures under the Wasserstein-Fisher-Rao geometry, and our solution to the second problem is inspired by the literature of consensus-based optimization. With this talk I hope to convey the multiple opportunities for mathematicians to participate in the conversation about pressing societal questions in the development of AI.

I will discuss the behavior of interaction energy minimizers on bounded domains. When the interaction potential is more singular than Newtonian, the mass does not tend to concentrate on the boundary; when it is Newtonian or less singular, the mass necessarily concentrates on the boundary for purely repulsive potentials. We also draw a connection between bounded-domain minimizers and whole-space minimizers.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We show that weak solutions of degenerate compressible Navier-Stokes equations converge to the strong solutions of the pressureless Euler system with linear drag term, Newtonian repulsion and quadratic confinement. The proof is based on the relative entropy method using the artificial velocity formulation for the one-dimensional Navier-Stokes system. The result is based on the joint work with Jose A. Carrillo and Ewelina Zatorska. Moreover we will shortly discuss how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler type hydrodynamic systems via the relative entropy method. This result is based on the joint work with Jose A. Carrillo and Yinping Peng.

In this talk, we will present a coupled Patlak-Keller-Segel-Navier-Stokes (PKS-NS) system that models chemotaxis phenomena in the fluid. The system exhibits critical threshold phenomena. For example, if the total population of the cell density is less than $8\pi$, then the solutions exist globally in time. Moreover, finite time blowup solutions exist if this population constraint is violated. We further show that globally regular solutions with arbitrary large cell populations exist. The primary blowup suppression mechanism is the shear flow mixing induced enhanced dissipation phenomena.

I will present the latest results and ideas related to the micro- to meso- and macroscopic limit for singular alignment dynamics. This includes the heterogeneous gradient flows related to weakly singular alignment (joint with David Poyato, University of Granada) with matrix valued communication, and a monokineticity estimate for strongly singular alignment (joint with Michał Fabisiak, University of Warsaw). In particular, I will show that any weakly continuous solution to strongly singular Cucker-Smale kinetic equation is monokinetic. This information can be used to obtain (via direct micro- to macroscopic mean-field limit) existence of measure-valued solutions to the fractional Euler-alignment system in the whole space for general initial data admitting vacuum.

In this talk, we discuss the existence theory for the Vlasov-alignment model with singular communication weights φ(r) = r −γ . In the case γ ∈ (0, d), we show the local-in-time existence of weak solutions, and the unique- ness is obtained for γ ∈ (0, d − 1]. We also consider the hypersingular com- munication weight, where γ ∈ (d, d + 1/4), and establish the local-in-time well-posedness for the Vlasov-alignment model .

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will present our recent results on the 1-dimensional hydrodynamic models of traffic, lubrication and collective behaviour in 1 dimension. I will discuss existence results, interesting two-velocity reformulations, singular limits (hard congestion) and long time behavior of solutions.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.