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Abstract: We will discuss a recent result joint with In-Jee Jeong and Theo Drivas. We prove that twisting in Hamiltonian flows on annular domains, which can be quantified by the differential winding of particles around the center of the domain, is stable to general perturbations. In fact, we prove the all-time stability of the lifted dynamics in an L2 sense (though single particle paths are generically unstable). These stability facts are used to establish several results related to the long-time behavior of inviscid fluid flows.

Abstract: The Enskog-Vlasov equation extends the Enskog equation for the dense hard sphere fluid by accounting for the attractive forces between, and the finite volume of, the gas particles. Hence, it gives a van-der-Waals-like description of a non-ideal gas, including liquid-vapor phase change. Specifically, the equation describes the liquid phase, the vapor phase, and a diffusive transition region connecting both phases. Solutions of the Enskog-Vlasov equation exhibit all relevant phenomena occurring in the evaporation and condensation of rarefied or dense vapors.

Using Grad’s moment method we derived macroscopic transport equations—moment equations with 13 and 26 variables—from the Enskog-Vlasov equation, which describe liquid vapor and transition region in terms of a few macroscopic properties.

Focussing on 1-D heat and mass transfer problems, we compare moment solutions to DSMC solutions for transport across the interface, and the interplay between interface and Knudsen layers. Interface resistivities for jump interface conditions are determined from the simulations, which show marked differences to those found from classical kinetic theory, where dimensionless resistivities are constants. In contrast, the EV models give temperature dependent resistivities, some negative off-diagonal resistivities, and indicate non-linear behavior where resistivities depend on mass and heat fluxes through the interface.

Abstract: We consider the COVID-19 pandemic in two Canadian regions: the Yukon Territory and the BC Interior. Our goal is to understand the effect of control measures and opinion dynamics on the transmission of the disease. We first incorporate an averaged measure of nine restriction policies - called the ‘Stringency Index’ - into the famous Susceptible-Infected-Recovered Model. We fit this model to case data for YT and BC Interior. To account for the effect of opinion changes, we extend the model splitting the susceptible compartment into four sub-compartments based on their view on vaccination and non-pharmaceutical interventions (NPIs). We also include vaccination though the vaccines are assumed to be 100% effective (no waning immunity). We fit this larger model to case data for each region and simulate the model results along with the Stringency Index to analyze the effect of policies and opinion dynamics.

Abstract: The epidemic curve is generally exponential during the early stage of an epidemic, presenting a challenge in identifying parameters of mathematical models that incorporate multiple control measures. This presents a major hurdle in disentangling and evaluating the effectiveness of contact tracing and other non-pharmaceutical public health interventions (NPIs) separately. In this presentation, we show how to use a novel contact tracing model and a simulation study to determine the dataset required for such an assessment. Our results show that the daily counts of new cases, cases diagnosed via contact tracing, and symptom onsets are necessary for this evaluation. We apply our method to the early stage of the Covid-19 pandemic in Ontario, Canada.

Abstract: In a linear population model that has a unique “largest” eigenvalue, the corresponding left and right (Perron) eigenvectors determine the long-term relative prevalence and reproductive value of different types of individuals, as described by the Perron-Frobenius theorem and generalizations. It is therefore of interest to study how the Perron vectors depend on the generator of the model. Even when the generator is a finite-dimensional matrix, there are several approaches to the corresponding perturbation theory. We explore an approach that hinges on stochasticization (re-weighting the space of types to make the generator stochastic) and interprets formulas in terms of the corresponding Markov chain. The resulting expressions have a simple form that can also be obtained by differentiating the renewal-theoretic formula for the Perron vectors. The theory appears well-suited to the study of infection spread that persists in a population at a relatively low prevalence over an extended period of time, via a fast-slow decomposition with the fast/slow variables corresponding to infected/non-infected compartments, respectively. This is joint work with MSc student Tareque Hossain.

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Abstract: We will discuss local well-posedness and ill-posedness results of some active scalar equations, including 2D incompressible Euler equations and the SQG equation. We will discuss how one can obtain instantaneous growth of solutions using a perturbation of a steady initial data as well as making use of unboundedness of the Riesz transform in $L^\infty$. We will then discuss the first result of an instantaneous gap loss of Sobolev regularity for 2D Euler. Namely, we will describe a construction of initial vorticity for the 2D Euler equations that belongs to the Sobolev space $H^\beta$, $\beta \in (0,1)$ which gives rise to a unique global-in-time solution that instantaneously leaves not only $H^\beta$, but also $H^{\beta'}$ for every $\beta' >(2-\beta )\beta /(2-\beta^2)$. This is joint work with Diego Cordoba and Luis Martinez-Zoroa.

Abstract:
In a recent article by Guglielmi and Hairer (SIADS 2015), an analysis in the $\varepsilon\to 0$ limit was proposed of regularized discontinuous ODEs in codimension-2 switching domains (intersections of discontinuity surfaces). This was obtained by studying a certain 2-dimensional system describing the so-called hidden dynamics. In particular, the existence of a unique limit solution was not proved in all cases, a few of which were labeled as ambiguous, and it was not clear whether or not the ambiguity could be resolved. We now show that it cannot be resolved in general. We show that three types of non-uniqueness or ambiguity can occur.

Firstly, we show that the limit solution can depend on the form of the regularization function.

Secondly, we show that behaviour of the hidden dynamics in the structurally ambiguous cases can depend on parameters, with bifurcations between different macroscopic outcomes. Thus the structure does not directly determine the behaviour.

Finally, we investigate the extreme sensitivity of solutions to initial conditions or parameters in the transition from codimension-2 domains to codimension-3 when there is a limit cycle in the hidden dynamics.

Abstract: We are interested in the design of forcing in the Navier–Stokes equation such that the resultant flow maximizes the heat transfer between two differentially heated walls for a given power supply budget. Previous work established that heat transport cannot scale faster than 1/3-power of the power supply. Recently, Tobasco and Doering (PRL'17) and Doering and Tobasco (CPAM'19) constructed self-similar two-dimensional steady branching flows, saturating this upper bound up to a logarithmic correction to scaling. We present a construction of three-dimensional ``branching pipe flows'' that eliminates the possibility of this logarithmic correction and for which the corresponding heat transport scales as a clean 1/3-power law in power supply. Our flows resemble previous numerical studies of the three-dimensional wall-to-wall problem by Motoki, Kawahara and Shimizu (JFM'18). However, using an unsteady branching flow construction, it appears that the 1/3 scaling is also optimal in two dimensions. After carefully examining these designs, we extract the underlying physical mechanism that makes the branching flows ``efficient,'' based on which we present a design of mechanical apparatus that, in principle, can achieve the best possible case scenario of heat transfer. We will further discuss some interesting implications of branching flows, for example, anomalous dissipation in turbulent flows and Rayleigh--B\'enard convection.