# Research Seminars: Applied and Computational Mathematics

## Fall 2017

Time & Location: Typically talks will be on Fridays in Gibson Hall 310 at 3:00 PM.
Organizers: Glatt-Holtz, Nathan and Zhao, Kun

February 23

## On the Geometric Regularity Criteria for Incompressible Navier--Stokes Equations

Siran LiRice University

Abstract:

We present some recent results on the regularity criteria for weak solutions to the incompressible Navier--Stokes equations (NSE) in 3 dimensions. By the work of Constantin--Fefferman, it is known that the alignment of vorticity directions is crucial to the regularity of NSE in $\R^3$.  In this talk we show a boundary  regularity theorem for NSE on curvilinear domains with oblique derivative boundary conditions. As an application, the boundary regularity of incompressible flows on balls, cylinders and half-spaces with Navier boundary condition is established, provided that the vorticity is coherently aligned up to the boundary. The effects of vorticity alignment on the $L^q$, $1<q<\infty$ norm of the vorticity will also be discussed.

March 2

## Regularity Criteria and the Dissipation Wave Number for Equations of Fluid Motion

Karen ZayaUniversity of Michigan

Abstract: The regularity of solutions to equations of fluid motion remains a significant open problem. A vast amount of literature has been devoted to studying regularity criteria, such as the classical Beale-Kato-Majda and Ladyzhenskaya-Prodi-Serrin regularity conditions. We will review some of this literature for the three-dimensional Euler, Navier-Stokes, Boussinesq, and magnetohydrodynamics equations. Then, in the framework of Kolmogorov’s theory of turbulence, we will discuss how work with the dissipation wavenumber and determining modes has produced new, weaker regularity criteria for these equations.

March 9

## Topic

SpeakerInstitution

Abstract: TBA

March 16

## Bridges Over Troubled Waters: How the WAVES Consortium is Using Science to Save Lives from Tsunami Hazards in Indonesia.

Ron HarrisBrigham Young University

Abstract:

Historical records we have compiled demonstrate that over the past 400 years there have been 105 tsunamis throughout Indonesia, which is an average of at least1 tsunami every 4 years.  Just in the past 22 years 8 tsunamis have struck Indonesia, 7 of which caused numerous fatalities (>200,000 total deaths). Women and children account for the majority of these deaths.  However, most of these lives could have been saved if those in harm’s way would have known they were at risk, the natural signs that a tsunami was approaching and how to respond.  These three fundamental aspects of tsunami disaster preparedness require an integrated, multidisciplinary approach to resolve as demonstrated in Japan during the 2011 tsunami that struck there.  The tsunamis in Indonesia and Japan were similar in size and impacted around the same numbers of people; yet, in Japan there were less than 1 death for every 10 in Indonesia.  The difference?  Resilience!

Building resilience to natural hazards in areas most at risk is the focus of the WAVES consortium, which is a multi-disciplinary partnership of academic and government research institutions dedicated to reversing the increasing losses to nature in developing countries vulnerable to natural hazards.  We identify those communities most at risk and help implement community-based disaster mitigation strategies that will save lives from future natural hazards.  The project has three integrated goals: 1) ‘Listen to Earth’ to determine who’s most at risk.  This task involves applying novel statistical inversion techniques to determine who’s most at risk based on historical, archeological and geological records of past hazardous events.  2) ‘Listen to the People’ through conducting questionnaire surveys to determine levels of awareness and readiness, and town hall meetings. This information is vital in designing presentations and public service videos to communicate risk and effective disaster mitigation strategies in a cultural context.  3) ‘Empower People to Listen to Earth’ by assisting local communities to implement and sustain their own risk reduction strategies connected to natural warning signs.

One of the most important technical aspects of the research is the construction of tsunami flooding maps for at risk communities.  These maps provided a way to communicate who is most at risk and the details for each site about expected tsunami escape times and run up heights.  A team that integrates the expertise of geoscientists, mathematicians and statisticians at BYU, Virgina Tech and Tulane University conducts the numerical modeling.

We used the tsunami inundation maps as a starting point to assist local communities in tsunami disaster mitigation planning and implementation of risk reduction strategies.  These maps include the areas likely to flood during a tsunami, predicted waves heights in these areas, the number of people inhabiting these areas and the time after the earthquake of the arrival of the tsunami.  Essentially, they communicate who is most at risk, safe evacuate sites and the time available to evacutate.  In most cases, those at risk can know a tsunami is approaching if they feel an earthquake that shakes for > 20 seconds.  At that point they may have only 20 minutes to escape to an elevation of 20 m.

Location:  Gibson Hall 310

Time 2:00

March 16

## Large Solutions of Compressible Euler Equations

Geng ChenUniversity of Kansas

Abstract:

Compressible Euler equations (introduced by Euler in 1757) model the motion of compressible inviscid fluids such as gases. It is well-known that solutions of compressible Euler equations often develop discontinuities, i.e. shock waves. Successful theories have been established in the past 150 years for small solutions in one space dimension. The theory on large solutions is widely open for a long time, even in one space dimension.  In this talk, I will discuss some recent exciting progresses in this direction. The talk is based on my joint works with A. Bressan, H.K. Jenssen, R. Pan, R. Young, Q. Zhang, and S. Zhu.

March 23

## Topic

SpeakerInstitution

Abstract: TBA

April 6

## The impact of laminar boundary layers on the search for the ultimate regime of turbulent convection

Abstract:

Using rigorous mathematical analysis, we demonstrate that the boundary layers in highly turbulent convection must remain laminar when the fluid satisfies a Navier-slip boundary condition, and the temperature field satisfies a fixed flux condition at the top and bottom plates.  Although this result is not surprising in the context of discussions of the ultimate regime of thermal convection, coupled with recent rigorous upper bounds for the same system, the presence of laminar boundary layers in this setting does raise questions on how the ultimate regime appears, and/or the efficacy of upper bound theory.

April 13

## Remarks on Onsager's Conjecture and Anomalous Dissipation on domains with and without boundaries.

Theo DrivasPrinceton University

Abstract:

We first discuss the inviscid limit of the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus.  Assuming that the solutions have Besov norms bounded uniformly in viscosity, we establish an upper bound on energy dissipation. As a consequence, Onsager-type "quasi-singularities" are required in the Leray solutions, even if the total energy dissipation is o(ν) in the limit ν → 0.  Next, we discuss an extension of Onsager's conjecture for domains with solid boundaries. We give a localized regularity condition for energy conservation of weak solutions of the Euler equations assuming Besov regularity of the velocity with σ>1/3 for any U⋐Ω and, on an arbitrary thin layer around the boundary, boundedness of velocity, pressure and continuity of the wall-normal velocity. We also prove that the global viscous dissipation vanishes in the inviscid limit for Leray-Hopf solutions of the Navier-Stokes equations under the similar assumptions, but holding uniformly in a vanishingly thin viscous boundary layer.  Finally, if a strong Euler solution exists, we show that equicontinuity at the boundary within a O(ν) strip alone suffices to conclude the absence of anomalous dissipation.

The first part of the talk concerns joint work with G. Eyink, the second with H.Q. Nguyen.

April 20

Abstract: TBA

April 23

## Topic

Shane McquarrieBrigham Young University

Abstract: TBA

April 27

## Modeling Evolution and Ecology of Heterogeneous Viral Strategies in Virus-Microbe Systems

Hayriye GulbudakUniversity of Louisiana at Lafayette (Host: Mac Hyman)

Abstract:

Viruses of microbes, including bacterial viruses (phage), archaeal viruses, and eukaryotic viruses, can influence the fate of individual microbes and entire populations. Here, we model distinct modes of virus-host interactions and study their impact on the abundance and diversity of both viruses and their microbial hosts. We consider two distinct viral populations infecting the same microbial population via two different strategies: lytic and chronic. A lytic strategy corresponds to viruses that exclusively infect and lyse their hosts to release new virions. A chronic strategy corresponds to viruses that infect hosts and then continually release new viruses via a budding process without cell lysis. The chronic virus can also be passed on to daughter cells during cell division. The long-term association of virus and microbe in the chronic mode drives differences in selective pressures with respect to the lytic mode. We utilize invasion analysis of the corresponding nonlinear differential equation model to study the ecology and evolution of heterogenous viral strategies. We first investigate stability of equilibria, and characterize oscillatory and bistable dynamics in some parameter regions. Then, we derive fitness quantities for both virus types and investigate conditions for competitive exclusion and coexistence.  In so doing we find unexpected results, including a regime in which the chronic virus requires the lytic virus for survival and invasion.

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu