PDE Seminar Spring 2024

Fridays, 3:30 — 4:20pm, Stevenson Center 1431 (in-person) or via Zoom (online). When possible, in-person talks will be live streamed.

Date: Friday, Jan 19th.

  • Speaker: Calum Rickard, UC Davis.
  • Title: An infinite class of shocks for compressible Euler
  • Abstract: We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Holder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Holder space is given in the form of a fractional series expansion.

Date: Friday, Feb 16th.

  • Speaker: Warren Li, Princeton University.
  • Title: Scattering towards the singularity in Kasner spacetimes
  • Abstract: The Kasner spacetimes are solutions to Einstein’s equations that play a decisive role in the study of spacelike singularities in GR. In this talk, we study two model hyperbolic systems in Kasner spacetimes, namely the scalar wave equation and the linearized Einstein(-scalar field) system, whose solutions have certain quantitative blow-up asymptotics near the singularity of Kasner. We present recent results deriving a sharp correspondence between Cauchy data and asymptotic data at the singularity, which reveals interesting phenomena regarding gains and losses of regularity.

Date: Friday, March 8th.

  • Speaker: Maxim Olshanskii, University of Houston.
  • Title: Finding equilibrium states of fluid membranes
  • Abstract:We are interested in finding equilibrium configurations of inextensible elastic membranes exhibiting lateral fluidity. Differential equations governing the mechanical equilibrium are derived using a continuum description of the membrane motions given by the surface Navier–Stokes equations with bending forces. Equilibrium conditions that are found appear to be independent of lateral viscosity and relate tension, pressure and tangential velocity of the fluid. These conditions yield that only surfaces with Killing vector fields, such as axisymmetric shapes, can support non-zero stationary flow of mass. We derive a shape equation that extends a classical Helfrich model with area constraint to membranes of non-negligible mass. We introduce a simple numerical method to compute solutions of this highly non-linear equation. The numerical method is then applied to find a diverse family of equilibrium configurations.

Date: Wednesday, March 20th. (Note the different date.)

  • Speaker: Yannis Angelopoulos, California Institute of Technology.
  • Title: TBA
  • Abstract: TBA

Date: Monday, April 1st. (Note the different date.)

  • Speaker: Bingyang Hu, Auburn University.
  • Title: TBA
  • Abstract: TBA

Date: Friday, April 5th.

Date: Friday, April 12th.

  • Speaker: Daniel Ginsberg, Brooklyn College — CUNY.
  • Title: TBA
  • Abstract: TBA