In August, Radu Dascaliuc (PI) was awarded National Science Foundation grant DMS-1516487 , “Collaborative research: Turbulent cascades and dissipation in the 3D Navier-Stokes model.” The main theme of the project is a rigorous study of various manifestations of turbulence in three-dimensional fluid flows modeled by the Navier-Stokes equations. This is considered both from the perspective of the mathematical theory of turbulence, and as a physical mechanism underlying possible blow-ups (singularities) of the solutions of the system. This is a three-year grant in the amount of $79,000.
National Science Foundation grant NSF DMS-1522734 "Phase transitions in porous media across multiple scales" was recently awarded to Prof. Malgorzata Peszynska (Principal Investigator). This is a three year grant 2015-18 in the amount $383,894K. Overview of the project is at http://www.nsf.gov/awardsearch/showAward?AWD_ID=1522734 . This grant in Computational Mathematics will involve graduate and undergraduate students and will combine modeling, rigorous analysis, and computations as well as continued collaborations with colleagues from oceanography, imaging science, and high performance computing.
A group with a cyclically symmetric presentation admits an automorphism of finite order called the shift. In this talk we look at cyclically presented groups which admit a certain decomposition, and relate the shift dynamics for the group to that of its decomposition. Topological methods are used to identify fixed points for powers of the shift.
I will talk about a probabilistic cascade structure that can be naturally associated with certain partial differential equations and how it can be used to study well-posedness questions.
In the context of the still unsolved uniqueness problem for the 3D Navier-Stokes equations, our aim will be to see how the explosion properties of such cascades help establish a connection between the uniqueness of symmetry-preserving (self-similar) solutions and the uniqueness of the general problem.
Based on the joint work with N. Michalowski, E. Thomann, and E. Waymire.
This is joint work with Rob Costa and Patrick Dynes. We prove that if an n x n matrix defined over the field of p-adic numbers satisfies a certain congruence property, then it has a strictly maximal eigenvalue in Q_p and that iteration of the (normalized) matrix converges to a projection operator onto the corresponding eigenspace. This result may be viewed as an analogue of the Perron-Frobenius theorem for positive real matrices.
Biofilm is a collection of microbial cells which stick together within a protective gel-like substance that they produce, and which adhere to a surface. In the talk we will report on recently published collaborative research in which we studied the evolution of biofilm in porous media.
The first step was to use X-ray microtomography images from the lab of Dorthe Wildenschild. The images inspired the construction of a new mathematical model for biofilm evolution, which uses a variational inequality to satisfy the maximum volume constraint observed by experimentalists. The mathematical model is a doubly nonlinear PDE system, and has solutions of very low regularity, thus it requires delicate numerical schemes. Further challenges include time-stepping but our computations are able to reproduce the biofilm morphologies similar to those seen in the images.
The images also provide the detailed geometry of the porous medium at porescale, and this geometry changes due to the biomass clogging the pores. We simulate these effects with a coupled hydrodynamics model which is in turn upscaled; the calculated conductivities compare very well to those obtained experimentally, at various flow rates. The modeling and computational efforts are joint work with Anna Trykozko (University of Warsaw). The project shares some similarities with hydrate modeling, and has been/is supported by NSF-DMS 1115827 and NSF-DMS 1522734.
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