PDE Seminar Spring 2017

Fridays, 4:10pm, Stevenson Center 1307

Date: Friday, January 13, 2017

  • Speaker: Ravshan Ashurov, Institute of Mathematics, National University of Uzbekistan
  • Title: On convergence almost everewhere of spectral resolutions of elliptic differential operators and the multiple Fourier integrals
  • Abstract:The question of the validity of the Luzin conjecture for the spherical partial sums of the multiple Fourier integrals is open so far. But if we consider the Riess means of the multiple Fourier integrals or in addition if we let f=0 on an open set G, and investigate the convergence to zero  a.e. on G (i.e. generalized localization principle), then there are many possitive results. We first remind some of these results, and then study generalized localization principle for compactly supported distributions and present sharp conditions for its fullfilment.

Date: Thursday, April 6, 2017

  • Speaker: Manuchehr Aminian, University of North Carolina Chapel Hill
  • Title: Skewness in the Passive Tracer Problem
  • Abstract: The broad goal of our research is to understand the statistics of a diffusing passive tracer under the influence of laminar pipe flow. Classical theory has shown that, on sufficiently long timescales, the tracer distribution is well modeled by an effective diffusion equation in the flow direction. However, the distribution is strongly asymmetric before this final diffusion timescale, and little work has previously been done to describe this, aside from the special cases of the circular pipe, and flow between infinite parallel plates. Here, we present our work studying this problem for different classes of cross-sectional geometries. We have uncovered a wealth of nontrivial sign dependence of the skewness depending on pipe shape, on both advective and diffusive timescales, and we see excellent agreement between theory, numerics, and experiments, all done here at UNC. This is joint work with Francesca Bernardi, Roberto Camassa, Daniel M. Harris, and Richard McLaughlin.
  • Date: Friday, April 14, 2017
    • Speaker: Edmond Rusjan, SUNY Polytechnic Institute
    • Title: A Mathematical Model of the Blood Flow in the Retina of the Eye
    • Abstract: Understanding the blood flow in the retina of the eye may provide insights in several eye pathologies and ultimately lead to better treatments. We model the flow as a generalized Darcy flow on a curved surface and solve the model numerically using discrete exterior calculus and finite element exterior calculus. Results support the hypothesis that changes in the shape of the retina cause significant changes in the ocular blood flow, which may play a role in the dynamics of open angle glaucoma.