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: Global constructions of impulsive gravitational wave spacetimes
  • Abstract: In this talk I will try to describe an upcoming result (which is joint work with Jonathan Luk) on the construction of global geodesically complete spacetimes that satisfy the Einstein vacuum equations and that contain two colliding impulsive gravitational waves. I will start by reviewing some of the classical results in the literature of the general theory of propagation of singularities for wave equations, then I will also review the important work of Luk-Rodnianski on local generic constructions, and the extensions of these results to the semi-global setting, before presenting the global theory. Finally, I will mention some open problems.

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

  • Speaker: Bingyang Hu, Auburn University.
  • Title: Suppression of Epitaxial Thin Film Growth by Mixing
  • Abstract: In this talk, we consider the fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus, with or without an advection term. We will show that in the absence of advection, there exists initial data which will make the solution blow up in finite time; while in the advective case, if the imposed advection is sufficiently mixing, the global existence of the solutions can be achieved. Finally, we will make some further remarks on the general framework on how advection can guarantee the global existence of certain non-linear equations. This talk is based on several joint work with Yu Feng, Xiaoqian Xu and Yeyu Zhang.

Date: Friday, April 5th.

  • Speaker: Arick Shao, Queen Mary University of London.
  • Title: Bulk-boundary correspondence for vacuum asymptotically Anti-de Sitter spacetimes
  • Abstract: The AdS/CFT conjecture in physics posits the existence of a correspondence between gravitational theories in asymptotically Anti-de Sitter (aAdS) spacetimes and field theories on their conformal boundary. In this presentation, we prove a rigorous mathematical statement toward this conjecture in the classical relativistic setting. In particular, we show there is a one-to-one correspondence between aAdS solutions of the Einstein-vacuum equations and a suitable space of data on the conformal boundary (consisting of the boundary metric and the boundary stress-energy tensor), provided the boundary satisfies a geometric condition. This is joint work with Gustav Holzegel, and builds upon joint works with Athanasios Chatzikaleas, Simon Guisset, and Alex McGill.
  • Zoom Link: https://vanderbilt.zoom.us/j/97163909916

Date: Friday, April 12th.

  • Speaker: Daniel Ginsberg, Brooklyn College — CUNY.
  • Title: The stability of irrotational shocks and the Landau law of decay
  • Abstract: It is well-known that in three space dimensions, smooth solutions to the equations describing a compressible gas can break down in finite time. One type of singularity which can arise is known as a “shock”, which is a hypersurface of discontinuity across which the integral forms of conservation of mass and momentum hold and through which there is nonzero mass flux. One can find approximate solutions to the equations of motion which describe expanding spherical shocks. We use these model solutions to construct global-in-time solutions to the irrotational compressible Euler equations with shocks. This is joint work with Igor Rodnianski.