• home

Abstract: I will discuss finite-time blowup for a model of the 3D Euler equation where the Helmholtz projection is replaced by a projection onto a more restrictive constraint space. The model preserves the structure of the self-advection nonlinearity entirely, and also conserves energy and helicity.

Abstract: The COVID-19 pandemic has placed an unprecedented strain on healthcare systems worldwide, making it difficult to accurately predict demand surge for acute care and allocate resources effectively. This paper proposes a novel predictive learning model that combines a differential equation model with delays and a multi-objective learning genetic algorithm to forecast hospitalizations and intensive care admissions based on predicted infections in a given population. Using data from Canada, Alberta, and Edmonton, we demonstrate the effectiveness of our model in significantly improving the ability to predict acute care capacity and manage resources during emergent infectious disease outbreaks such as COVID-19. Our approach is tailored to different population segments and considers various factors, such as variant strains of the virus and vaccination campaigns. Our model represents a significant contribution to the field, providing healthcare managers and decision-makers with a powerful tool to plan resources in the face of epidemics and major disasters. We contribute to the conversation on how we respond to public health emergencies, enabling more accurate predictions of demand for care and allocation of resources, ultimately saving lives and limiting the impact of these crises on our communities.

This is a joint work with Y. Bahri, N. Gilani, S. Ibrahim, A. Park and O. Chakroun

Abstract: One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete, which is a significant difficulty when modeling chaotic systems whose solutions are sensitive to initial conditions. If one instead has snapshots of a system, i.e. data, one can make a better guess at the true state by incorporating the data via data assimilation. Many of the most popular data assimilation methods were developed for general physical systems, not just turbulent flows or chaotic systems. However, in the context of fluids, data assimilation works better than would be anticipated for a general physical system. In particular, turbulent fluid flows have been proven to have the property that, given enough perfect observations, one can recover the full state irrespective of the choice of initial condition. This property is unique to turbulent fluid flows, a consequence of their finite dimensionality. In this presentation, we will discuss the continuous data assimilation algorithm that was used to prove the convergence in the original, perfect data setting, present various robustness results of the continuous data assimilation algorithm, and discuss how continuous data assimilation can be used to identify and correct model error. Moreover, we will highlight the efforts of our other current research into long-standing problems in the stability of fluid flows, and how we have discovered some very interesting connections to our data assimilation research.

The aim of this talk is to present some theoretical and numerical results concerning Beltrami fields, called force-free fields in solar and fusion physics. The talk will be divided into three parts. In the first part, we will recall the physical context of the appearance of these fields, i .e. essentially the modeling of magnetic equilibria in solar physics. In the second part, we will recall some mathematical theoretical results concerning the existence and properties of fields in any three-dimensional domain. In the third and last part, we will expose some methods to compute these 3D fields and the associated physical equilibria, as well as some numerical results obtained with our codes.

Abstract: This talk is devoted to the Relativistic Vlasov-Maxwell system in space dimension three. We prove the local smooth solvability for weak topologies. This result is derived from a representation formula informing about the momentum spread, showing that the domain of influence in momentum is controlled by mild information in the sense that derivatives are not (or less than before) implied. In doing so, we develop a Radon Fourier analysis on the RVM system, which leads to the study of a class of singular weighted integrals. The end of my talk, I show how this method is applied in order to construct smooth solutions to the RVM system in the regime of hot, dense and strongly magnetized plasmas. This is a joint work with C. Cheverrey.

Abstract: Verifying nonlinear stability of a laminar fluid flow against all perturbations is a classic challenge in fluid dynamics. All past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. This "energy method" cannot show global stability of any flow in which perturbation energy may grow transiently. For the many flows that allow transient energy growth but seem to be globally stable (e.g. pipe flow and other parallel shear flows at certain Reynolds numbers) there has been no way to mathematically verify global stability. After explaining why the energy method was the only way to verify global stability of fluid flows for over 100 years, I will describe a different approach that is broadly applicable but more technical. This approach, proposed in 2012 by Goulart and Chernyshenko, uses sum-of-squares polynomials to computationally construct non-quadratic Lyapunov functions that decrease monotonically for all flow perturbations. I will present a computational implementation of this approach for the example of 2D plane Couette flow, where we have verified global stability at Reynolds numbers above the energy stability threshold. This energy stability result for 2D Couette flow had not been improved upon since being found by Orr in 1907. The results I will present are the first verification of global stability – for any fluid flow – that surpasses the energy method. This is joint work with Federico Fuentes (Universidad Católica de Chile) and Sergei Chernyshenko (Imperial College London).

Abstract: Schrödinger operators with delta potentials are of longstanding interest for admitting closed analytic solutions, and they receive renewed attention for connections to the Kardar–Parisi–Zhang equation. Along with a review of this background, the talk will discuss a standard model of the operators in 2D and its counterpart in Feynman–Kac formulas.