Existence and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system over moving domains

This dissertation concerns the well-posedness of the Navier-Stokes-Smoluchowski system. The system models a mixture of fluid and particles in the so-called bubbling regime. The compressible Navier-Stokes equations governing the evolution of the fluid are coupled to the Smoluchowski equation for the particle density at a continuum level. First, working on fixed domains, the existence of weak solutions is established using a three-level approximation scheme and based largely on the Lions-Feireisl theory of compressible fluids. The system is then posed over a moving domain. By utilizing a Brinkman-type penalization as well as penalization of the viscosity, the existence of weak solutions of the Navier-Stokes-Smoluchowski system is proved over moving domains. As a corollary the convergence of the Brinkman penalization is proved. Finally, a suitable relative entropy is defined. This relative entropy is used to establish a weak-strong uniqueness result for the Navier-Stokes-Smoluchowski system over moving domains, ensuring that strong solutions are unique in the class of weak solutions.

Fundamental domains for proper affine actions of Coxeter groups in three dimensions

We study proper actions of groups $G \cong \Z/2\Z \ast \Z/2\Z \ast \Z/2\Z$ on affine space of three real dimensions. Since $G$ is nonsolvable, work of Fried and Goldman implies that it preserves a Lorentzian metric. A subgroup $\Gamma < G$ of index two acts freely, and $\R^3/\Gamma$ is a Margulis spacetime associated to a hyperbolic surface $\Sigma$. When $\Sigma$ is convex cocompact, work of Danciger, Gu{\'e}ritaud, and Kassel shows that the action of $\Gamma$ admits a polyhedral fundamental domain bounded by crooked planes. We consider under what circumstances the action of $G$ also admits a crooked fundamental domain. We show that it is possible to construct actions of $G$ that fail to admit crooked fundamental domains exactly when the extended mapping class group of $\Sigma$ fails to act transitively on the top-dimensional simplices of the arc complex of $\Sigma$. We also provide explicit descriptions of the moduli space of $G$ actions that admit crooked fundamental domains.

Flooded with Information from Social Media: Effects of Disaster Information Source and Visuals on Viewers' Cognitive, Affective, and Behavioral Responses

While a variety of crisis types loom as real risks for organizations and communities, and the media landscape continues to evolve, research is needed to help explain and predict how people respond to various kinds of crisis and disaster information. For example, despite the rising prevalence of digital and mobile media centered on still and moving visuals, and stark increases in Americans’ use of visual-based platforms for seeking and sharing disaster information, relatively little is known about how the presence or absence of disaster visuals online might prompt or deter resilience-related feelings, thoughts, and/or behaviors. Yet, with such insights, governmental and other organizational entities as well as communities themselves may best help individuals and communities prepare for, cope with, and recover from adverse events. Thus, this work uses the theoretical lens of the social-mediated crisis communication model (SMCC) coupled with the limited capacity model of motivated mediated message processing (LC4MP) to explore effects of disaster information source and visuals on viewers’ resilience-related responses to an extreme flooding scenario. Results from two experiments are reported. First a preliminary 2 (disaster information source: organization/US National Weather Service vs. news media/USA Today) x 2 (disaster visuals: no visual podcast vs. moving visual video) factorial between-subjects online experiment with a convenience sample of university students probes effects of crisis source and visuals on a variety of cognitive, affective, and behavioral outcomes. A second between-subjects online experiment manipulating still and moving visual pace in online videos (no visual vs. still, slow-pace visual vs. still, medium-pace visual vs. still, fast-pace visual vs. moving, slow-pace visual vs. moving, medium-pace visual vs. moving, fast-pace visual) with a convenience sample recruited from Amazon’s Mechanical Turk (mTurk) similarly probes a variety of potentially resilience-related cognitive, affective, and behavioral outcomes. The role of biological sex as a quasi-experimental variable is also investigated in both studies. Various implications for community resilience and recommendations for risk and disaster communicators are explored. Implications for theory building and future research are also examined. Resulting modifications of the SMCC model (i.e., removing “message strategy” and adding the new category of “message content elements” under organizational considerations) are proposed.