Vortex filament solutions of the Navier-Stokes equations

We consider solutions of the Navier-Stokes equations in $3d$ with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of $3d$ Navier-Stokes, as well as solutions which are locally approximately self-similar.Comment: 89 page

Finite depth gravity water waves in holomorphic coordinates

In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match the earlier infinite depth result of Hunter-Ifrim-Tataru.Comment: 82 pages, 1 figur

The lifespan of small data solutions to the KP-I

We show that for small, localized initial data there exists a global solution to the KP-I equation in a Galilean-invariant space using the method of testing by wave packets.Comment: 20 pages, minor update

Opposing effects of reward and punishment on human vigor

The vigor with which humans and animals engage in a task is often a determinant of the likelihood of the task’s success. An influential theoretical model suggests that the speed and rate at which responses are made should depend on the availability of rewards and punishments. While vigor facilitates the gathering of rewards in a bountiful environment, there is an incentive to slow down when punishments are forthcoming so as to decrease the rate of punishments, in conflict with the urge to perform fast to escape punishment. Previous experiments confirmed the former, leaving the latter unanswered. We tested the influence of punishment in an experiment involving economic incentives and contrasted this with reward related behavior on the same task. We found that behavior corresponded with the theoretical model; while instantaneous threat of punishment caused subjects to increase the vigor of their response, subjects’ response times would slow as the overall rate of punishment increased. We quantitatively show that this is in direct contrast to increases in vigor in the face of increased overall reward rates. These results highlight the opposed effects of rewards and punishments and provide further evidence for their roles in the variety of types of human decisions

A comparative study of some models of incoherence at the mesoscopic scale

The pre-existing literature on phenomena at the mesoscopic scale is concerned among other things with phase coherent transport. Phase coherent transport dominates at very low temperatures. With increase in temperature, as the system size becomes comparable to the inelastic mean free path phase incoherence sets in. This incoherence further leads to dephasing, and as a consequence purely quantum effects in electron transport give way to classical macroscopic behavior. In this work we consider two distinct phenomenological models of incoherent transport, the Coherent Absorption and Wave Attenuation models. We reveal some physical problems in the Coherent Absorption model as opposed to the Wave Attenuation model. We also compare our proposed model with experiments in case of the much studied peak to valley ratios in resonant tunneling diodes, magneto-conductance oscillations and Fano resonances in case of Aharonov-Bohm rings.Comment: 20 pages, 9 figure