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ACMS Abstracts: Spring 2016

Stefan Llewellyn Smith (UCSD)

Hollow vortices

Hollow vortices are vortices whose interior is at rest. They posses vortex sheets on their boundaries and can be viewed as a desingularization of point vortices. After giving a history of point vortices, we obtain exact solutions for hollow vortices in linear and nonlinear strain and examine the properties of streets of hollow vortices. The former can be viewed as a canonical example of a hollow vortex in an arbitrary flow, and its stability properties depend on a single non-dimensional parameter. In the latter case, we reexamine the hollow vortex street of Baker, Saffman and Sheffield and examine its stability to arbitrary disturbances, and then investigate the double hollow vortex street. Implications and extensions of this work are discussed.

Tom Solomon (Bucknell)

Experimental studies of reaction front barriers in laminar flows

We present studies of the effects of vortex-dominated fluid flows on the motion of reaction fronts produced by the excitable Belousov-Zhabotinsky reaction. The results of these experiments have applications for advection-reaction-diffusion dynamics in a wide range of systems including microfluidic chemical reactors, cellular-scale processes in biological systems, and blooms of phytoplankton in the oceans. To predict the behavior of reaction fronts, we adapt tools used to describe passive mixing.In particular, the concept of an invariant manifold is extended to account for reactive burning. Burning invariant manifolds (BIMs) are predicted as one-way barriers that locally block the motion of reaction fronts. These ideas are tested and illustrated experimentally in a chain of alternating vortices, a spatially-random flow, vortex flows with imposed winds, and a three-dimensional, nested vortex flow. We also discuss the applicability of BIM theory to the motion of bacteria in fluid flows.

Daniele Cappelletti (KU)

Deterministic and stochastic reaction networks

Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development.

Lihui Chai (UCSB)

Semiclassical limit of the Schrödinger-Poisson-Landau-Lifshitz-Gilbert system

The Schrödinger-Poisson-Landau-Lifshitz-Gilbert (SPLLG) system is an effective microscopic model that describes the coupling between conduction electron spins and the magnetization in ferromagnetic materials. This system has been used in connection to the study of spin transfer and magnetization reversal in ferromagnetic materials. In this paper, we rigorously derive the Vlasov-Poisson-Landau-Lifshitz-Glibert system as the semiclassical limit of SPLLG. The major difficulties come from the presence of the spin-magnetization coupling and the discontinuities of the magnetization at the boundary of the material. To overcome these difficulties, we first take the semiclassical limit (vanishing Planck constant) of a smoothed SPLLG system, and then the limit of vanishing smoothing parameter. As a byproduct, we prove the local existence and uniqueness of classical solutions to the smoothed SPLLG system.

Alejandro Roldan-Alzate (UW)

Non–invasive patient-specific cardiovascular fluid dynamics

Comprehensive characterization and quantification of blood flow is essential for understanding the function of the cardiovascular system under normal and diseased conditions. This provides important information not only for the diagnosis and treatment planning of different cardiovascular diseases but also for the design of cardiovascular devices. However, the anatomical complexity and multidirectional nature of physiological and pathological hemodynamics makes non-invasive characterization and quantification of blood flow difficult and challenging. Doppler ultrasound, a standard imaging technique, is limited to providing information on large vessels and calculating instantaneous average flow within the cardiac cycle. Magnetic resonance imaging (MRI) is increasingly being used for fluid dynamics analyses of cardiovascular diseases, including pulmonary arterial hypertension, portal hypertension and congenital heart diseases. Although two-dimensional (2D) phase contrast (PC) magnetic resonance imaging (MRI) measures velocity across a plane, it is still limited in its ability to fully characterize these complex flow systems. Four- dimensional (4D) flow MRI obtains velocity measurements in three dimensions throughout the entire cardiac cycle. Several attempts have been made to non-invasively characterize the blood flow dynamics of different cardiovascular diseases using the combination of medical imaging and computational fluid dynamics modeling (CFD). Idealized geometries, as well as patient-specific anatomies, have been used for computational simulations, which have improved the understanding of the fluid dynamics phenomena in different vascular territories. While CFD modeling can provide powerful insights and the potential for simulating different physiological and pathological conditions in the cardiovascular system, it is currently not reliable for use in clinical care. Based on different studies, additional work is needed to verify the accuracy of current CFD approaches or identify and address current shortcomings. The overall purpose of this research is to develop, implement and validate non-invasive flow analysis methodologies to assess cardiovascular flow dynamics, using a combination of 4D flow MRI, numerical simulations and patient-specific physical models. In this seminar, multidisciplinary work will be presented first, where different cardiovascular pathologies have been studied, such as congenital heart disease and portal hypertension using in vivo, in vitro and computational models. Second, some advances will be presented and a future outlook into the valuable contribution of engineering in the medical imaging and diagnostic technology will be provided.

Yachun Li (SJTU)

Vanishing viscosity limit of the compressible Isentropic Navier-Stokes equations with degenerate viscosities

In this talk we first establish the local-in-time well-posedness of the unique regular solution to the compressible isentropic Navier-Stokes equations with density-dependent viscosities in a power law and with vacuum appearing in some open set or at the far field, then after establishing uniform energy-type estimates with respect to the viscosity coefficients for the regular solutions we prove the convergence of the regular solution of the Navier-Stokes equations to that of the Euler equations with arbitrarily large data containing vacuum.

Peter Hinow (UW-Milwaukee)

Aspects of mathematical modeling of drug delivery

There are a variety of devices for the delivery of pharmaceutical substances, tablets of course being the most prominent. Pharmaceutical scientists and physicians have formulated goals, such as release of a drug in a controlled fashion over an extended period of time or the targeted delivery of a drug to a specific site in a patient’s body. Since experiments with these delivery devices can be costly and sometimes only partially conclusive, mathematical modeling plays a considerable role in understanding the mechanisms behind experimental release profiles, and in developing delivery systems. We present our works on drug delivery by matrix tablets and targeted drug delivery to the brain. This is joint work with Ami Radunskaya (Department of Mathematics, Pomona College, Claremont, CA) and Ian Tucker (School of Pharmacy, University of Otago, Dunedin, New Zealand), and has been supported by NSF grants DMS 1016214 and DMS 1016136 .

Jörn Dunkel (MIT)

Pattern formation in soft and biological matter

Identifying the generic ordering principles that govern multicellular and intracellular dynamics is essential for separating universal from system-specific aspects in the physics of living organisms. In this talk, we will survey and compare three recently proposed nonlinear continuum theories, which aim to describe pattern formation and topological defect structures in soft elastic bilayer materials, dense bacterial suspensions and ATP-driven active liquid crystals. We will discuss the phase diagrams of the three models, relate their predictions to experiments, and emphasize the underlying universality ideas. The good agreement with experimental data supports the idea that non-equilibrium pattern formation in a broad range of soft and active matter systems can be described effectively within the same class of higher-order partial differential equations.

Kalin Vetsigian (UW)

Interactions and dynamics in communities of antibiotic producing bacteria

How microbial diversity is generated and maintained is a fundamental ecological question that can be studied through laboratory microcosm experiments and mathematical modeling. We developed a platform for measuring interactions and dynamics in communities of antibiotic producing bacteria and examined patterns of microbial interactions and frequency-dependent selection. We discovered high levels of antibiotic inhibition, high levels of antibiotic degradation, and frequent bistability among pairs of strains. This data led to the theoretical realization that the interplay between antibiotic production and degradation can lead to robust diversity maintenance through simple motifs that contain bistable pairs. Further computer simulations showed that evolution of antibiotic production and degradation capabilities can lead to spontaneous emergence of diversity and complex eco-evolutionary dynamics in simple environments with a single food source.

Atsushi Mochizuki (RIKEN, Japan)

Dynamics of complex biological systems determined/controlled by minimal subsets of molecules in regulatory networks

Modern biology provides many networks describing regulations between a large number of species of molecules. It is widely believed that the dynamics of molecular activities based on such regulatory networks are the origin of biological functions. In this study we develop a new theory to provide an important aspect of dynamics from information of regulatory linkages alone. We show that the "feedback vertex set" (FVS) of a regulatory network is a set of "determining nodes" of the dynamics. It assures that (i) any long-term dynamical behavior of the whole system, such as steady states, periodic oscillations or quasi-periodic oscillations, can be identified by measurements of a subset of molecules in the network, and that (ii) the subset is determined from the regulatory linkage alone. For example, dynamical attractors possibly generated by a signal transduction network with 113 molecules can be identified by measurement of the activity of only 5 molecules, if the information on the network structure is correct. We also demonstrate that controlling the dynamics of the FVS is sufficient to switch the dynamics of the whole system from one attractor to others, distinct from the original.

Further Reading:

Mochizuki A., Fiedler B., Kurosawa G. and Saito D. (2013) Dynamics and control at feedback vertex sets. J. Theor. Biol. 335: 130-146.

Igor Mezic (UC Santa Barbara)

Koopman Operator Theory in Fluid Mechanics

There is long history of contributions to modal representation of nonlinear fluid flows, including global modes, triple decomposition and Proper Orthogonal Decomposition. Recently, a spectral decomposition relying on Koopman operator theory has attracted interest in the fluid mechanics community. Part of the attractiveness of the Koopman operator approach stems from ability to compute the modes from relatively simple algorithms such as the Dynamic Mode Decomposition (DMD). To make a connection between the two, we show an explicit relationship between a basic version of the Dynamic Mode Decomposition (DMD) and the Koopman Mode Decomposition (KMD) of dynamical systems, that allows for estimates of validity of approximation of Koopman modes by DMD modes, and in the process introduce the notion of Generalized Laplace analysis that enables the underlying calculations. We also discuss the notion of exact solutions and Resolvent Mode Decomposition in connection with Koopman Modes. In addition, traditionally the accuracy of decomposition methods has been tested by comparing the flow with its modal approximation in quadratic norm sense. We will discuss an alternative approach that incorporates recent advances in characterizing kinematics of aperiodic-in-time flows on finite time-scales by using the concept of mesohyperbolicity. This coupling of Koopman Mode Decomposition and Mesohyperbolicity Theory enables us to start utilizing dynamical systems theories of mixing in the context of flows with non-periodic time dependence.

Sushmita Roy (UW)

Clustering and graph learning to decipher gene regulatory networks on hierarchically related biological contexts

Gene regulatory networks are networks of genes and transcription factor proteins that constitute the core information processing machinery in living cells. Identifying the structure and how they determine the overall state of a cell is one of the major challenges in systems biology. However, inference of regulatory networks is a computationally and experimentally difficult problem. In this talk, I will present some of our recent work in developing machine learning based algorithms to infer regulatory networks from observed gene expression levels while imposing additional constraints on the inferred networks. Our work relies on some key structural and organizational properties of regulatory networks, namely that regulatory networks exhibit a modular structure, and often share components with an ancestral network. I will present results of applying our approaches to diverse types of species including unicellular yeasts to multi-cellular fish and mammalian species. Our computational methods can be used to systematically identify regulatory networks and how they change across different temporally and hierarchically related contexts such as cell types on a cell lineage or species in a phylogeny.

Randall Kamien (U Penn)

Liquid crystals and their (algebraic) topology

Liquid Crystals, the materials in your iPhone, are complex materials with varying degrees of internal order. I will discuss and demonstrate how algebraic topology can be used to identify and characterize long-lived configurations. I will also describe how conic sections naturally arise in these structures as intersections of simple polynomials.

Eleuterio Toro (U Trento)

A flux splitting approach to a class of hyperbolic systems

This talk is based on a recently proposed flux vector splitting method for the Euler equations which, compared to existing splitting methods, has some distinctive advantages, such as exact recognition of stationary isolated contact/shear waves, simplicity, robustness and efficiency. Distinguishing features of the new flux splitting approach are: complete separation of pressure from advection terms and identification of a reduced pressure system that furnishes all required information for constructing the full numerical flux in a simple manner. The resulting first-order method was originally proposed for the 1D Euler equations for ideal gases. In this talk I will first present the scheme as applied to the 3D Euler equations with general equation of state. Then I shall describe its extension to high-order of accuracy in both space and time, on unstructured meshes, using the ADER approach. Performance of the resulting method is illustrated through some carefully chosen test problems. I finish this presentation by mentioning extensions of this novel flux splitting approach to other hyperbolic systems, such as the MHD equations and the Baer-Nunziato equations for compressible multiphase flows.

Clotilde Fermanian Kammerer (Insmi - CNRS)

A kinetic model for the transport of electrons in graphene

This talk is based on a joint work with Florian Méhats (Université de Rennes 1) where we propose a new numerical model for computation of the transport of electrons in a graphene device. The underlying quantum model for graphene is a massless Dirac equation, whose eigenvalues display a conical singularity responsible for non adiabatic transitions between the two modes. The derived kinetic model takes the form of two Boltzmann equations coupled by a collision operator modeling the non adiabatic transitions. We shall discuss the collision term which includes a Landau-Zener transfer term and a jump operator, and describe algorithmic realizations of the semi-group solving this kinetic model by a particle method.

Annalisa Calini (College of Charleston)

"Stability of Vortex Filaments"

The Vortex Filament Equation or VFE, a model for the self-induced motion of a vortex filament in an ideal fluid, is a simple but important example of integrable geometric evolution equation for space curves. Its connection with the cubing focusing Nonlinear Schrödinger equation through the well-known Hasimoto transformation allows the use of many of the tools of soliton theory to construct and investigate finite-gap and soliton solutions. I will discuss linear and nonlinear stability properties of some of these solutions, including filaments in the shape of torus knots and solitons on vortex filaments. This is joint work with with Tom Ivey and Stéphane Lafortune.