PDE Seminar Fall 2021

Fridays, 2:30 — 3:30pm, Stevenson Center 1312 (in-person) or via Zoom (online).

Date: Friday, Sept. 3rd, 2021.

  • Speaker: Hengrong Du, Vanderbilt University.
  • Title: On hydrodynamics of nematic liquid crystals.
  • Abstract: In this talk, we will report some recent developments on analytic issues on two model PDE systems of hydrodynamics of nematic liquid crystals: the Ericksen—Leslie (EL) system and the Beris—Edwards (BE) system. They are both strongly coupled PDE systems between incompressible Navier–Stokes equations for the background fluid velocity field and gradient-flow-like equations for the order parameter field describing the averaged alignment of liquid crystal molecules. We will include (i) a new concentrated-compactness in 2-D for simplified EL system; (ii) existence and partial regularity in 3-D for co-rotational BE system and general EL with Ginzburg—Landau approximation. This is based on joint works with Tao Huang (Wayne State) and Changyou Wang (Purdue).

Date: Friday, Sept. 10th, 2021.

  • Speaker: Xinyue Zhao, Vanderbilt University.
  • Title: A free boundary tumor growth model with a time delay in cell proliferation.
  • Abstract: Being a leading cause of death, tumor is one of the most important health
    problems facing the whole world. While there is a lot of work on the tumor
    growth models, only a few of them included time delay; and nearly in all the
    literature, only the radially symmetric case was considered with a time delay.
    In this talk, I will present a non-radially symmetric tumor growth model
    with a time delay in cell proliferation. The time delay represents the time taken
    for cells to undergo cell replication (approximately 24 hours). The model is a
    coupled system of an elliptic equation, a parabolic equation and a backward or-
    dinary dierential equation. It incorporates the cell location under the presence
    of time delay, with the tumor boundary as a free boundary. The inclusion of a
    small time delay makes the system non-local, which produces technical dicul-
    ties for the PDE estimates. I will discuss the stability and bifurcation results we
    obtained concerning this model. Through stability analysis, the result indicates
    that tumor with large aggressiveness parameter would trigger instability, which
    is biologically reasonable..

Date: Friday, Oct. 8th, 2021.

  • Speaker: Elena Giorgi, Columbia University.
  • Title: The stability of charged black holes.
  • Abstract: Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.
  • Zoom link: https://vanderbilt.zoom.us/j/94962549783

Date: Friday, Oct. 22nd, 2021.

  • Speaker: Greg Fournodavlos, Princeton University.
  • Title: The mysterious nature of the big bang singularity
    • Abstract: 100 years ago, Kasner and Friedmann published their (separate) pioneering works on the discovery of the first exact cosmological solutions to Einstein’s field equations, revealing the presence of a striking new phenomenon, namely, the Big Bang singularity. Since then, it has been the object of study in a great deal of research on general relativity. However, the nature of the `generic’ Big Bang singularity still remains a mystery. Rivaling scenarios are abound (monotonicity, chaos, spikes) that make the classification of all solutions a very intricate problem. I will give a historic overview of the subject and describe recent advancements that confirm a small part of the conjectural picture.
    • This talk will take at a different time and place than our usual seminars: 1 pm, Sarratt 189.

Date: Friday, Oct. 29th, 2021.

  • Speaker: Katrina Morgan, Northwestern University.
  • Title: TBA
  • Abstract: TBA

Date: Friday, Nov. 5th, 2021.

  • Speaker: Junyan Zhang, Johns Hopkins University.
  • Title: TBA
  • Abstract: TBA

Date: Friday, Nov. 12th, 2021.

  • Speaker: Federico Pasqualotto, Duke University.
  • Title: TBA
  • Abstract: TBA

Date: Friday, Dec. 3rd, 2021.

  • Speaker: Ákos Nagy, UC Santa Barbara.
  • Title: TBA
  • Abstract: TBA

Date: Friday, Dec. 10th, 2021.

  • Speaker: Maxime Van de Moortel, Princeton University.
  • Title: TBA
  • Abstract: TBA