PDE Seminar Spring 2023

Fridays, 3:30 — 4:20pm, Stevenson Center 1310 (in-person) or via Zoom (online). When possible, in-person talks will be live streamed.

Date: Friday, February 3rd, 2023.

  • Speaker: Sanchit Chaturvedi, Stanford University.
  • Title: Vanishing viscosity and shock formation in Burgers equation
  • Abstract: I will talk about the shock formation problem for 1D Burgers equation in the presence of small viscosity. Although the vanishing viscosity problem till moments before the first shock and in presence of a fully developed shocks is very classical, little is known about the moment of shock formation. We develop a matched asymptotic expansion to describe the solution to the viscous Burgers equation (with small viscosity) to arbitrary order up to the first singularity time. The main feature of the work is the inner expansion that accommodates the viscous effects close to the shock location and match it to the usual outer expansion (in viscosity). We do not use the Cole-Hopf transform and hence we believe that this approach works for more general scalar 1D conservation laws. Time permitting, I will talk about generalizing to vanishing viscosity limit from compressible Navier–Stokes to compressible Euler equations. This is joint work with Cole Graham (Brown university).

Date: Friday, February 10th, 2023.

  • Speaker: Ryan Unger, Princeton University.
  • Title: Retiring the third law of black hole thermodynamics
  • Abstract: In this talk I will present a rigorous construction of examples of black hole formation which are exactly isometric to extremal Reissner–Nordström after finite time. In particular, our result can be viewed as a definitive disproof of the “third law of black hole thermodynamics.” This is joint work with Christoph Kehle.

Date: Friday, February 17th, 2023.

  • Speaker: Ronghua Pan, Georgia Institute of Technology.
  • Title: Isentropic Approximation
  • Abstract: In the study of compressible flows, the isentropic model was often used to replace the more complicated full system when the entropy is near a constant. This is based on the expectation that the corresponding isentropic model is a good approximation to the full system when the entropy is sufficiently close to the constant. We will discuss the mathematical justification of isentropic approximation in Euler flows and in Navier-Stokes-Fourier flows. This is based on the joint work with Y. Chen, J. Jia, and L. Tong.

Date: Friday, February 24th, 2023.

  • Speaker: Christian Gesse, Heinrich Heine University Dusseldorf, Germany.
  • Title: Long-term Dynamics for a Living Fluid Model
  • Abstract: We consider generalized Navier Stokes equations in the periodic setting to describe the dynamics of highly concentrated bacterial suspensions in n=2 and n=3 dimensions. First, we consider stability and instability of the ordered polar states of the system, which form a manifold of equilibria. In the setting of infinite dimensional dynamical systems, we show the existence of absorbing sets of arbitrary high regularity. This leads to the existence of a global attractor that determines the long-term dynamics of the system.Joint work with Christiane Bui and Jürgen Saal.

Date: Monday, February 27th, 2023. 

  • Speaker: Jeremy LeCrone, University of Richmond
  • Room change: SC 1320
  • Title: Kinodynamic Control Systems and Discontinuities in Clearance
  • Abstract: Control theory studies the evolution of dynamical systems which are actively influenced by some external agent (or controller). We will discuss control systems with kinodynamic constraints on admissible trajectories, wherein one encounters obstacles in state space which must be avoided as the system evolves. In this setting, one defines a system-dependent clearance function quantifying the shortest admissible distance to the obstacle set. We will focus on points of discontinuity in the clearance function and how these discontinuities are experienced as one traverses admissible trajectories, both away from and close to the obstacle set. Further, we will discuss settings in which specific discontinuities arise on the boundary of an obstacle which generate a structure of discontinuities propagating out into free space. A number of real–world systems will be introduced and discussed throughout the talk. The contents of the talk should be accessible to anyone with a basic understanding of differential equations and analysis, no previous knowledge of control theory will be expected.

Date: Friday, March 3rd, 2023.

  • Speaker: Allen Fang, Princeton University.
  • Title: A new proof for the nonlinear stability of slowly-rotating Kerr-de Sitter
  • Abstract: The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof uses vectorfield techniques to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability.

Date: Friday, March 31st, 2023.

  • Speaker: Shi-Zhuo Looi, University of Kentucky.
  • Title: Asymptotics for odd- and even-dimensional waves.
  • Abstract: In this talk, I will give a survey of recent and upcoming results on various linear, semilinear and quasilinear wave equations on a wide class of dynamical spacetimes in various even and odd spatial dimensions. These results include asymptotics for a wide range of nonlinearities. For many of these results, the spacetimes under consideration have only weak asymptotic flatness conditions and are allowed to be large perturbations of the Minkowski spacetime, provided that an integrated local energy decay estimate holds. We explain the dichotomy between even- and odd-dimensional wave behaviour. Part of this work is joint with Mihai Tohaneanu and Jared Wunsch.

Date: Friday, April 7th, 2023.

  • Speaker: Lili He, Johns Hopkins University.
  • Title: The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes.
  • Abstract: I will discuss the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. The proof uses tools from microlocal analysis and a detailed description of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator at low frequencies.

Date: Friday, April 14th, 2023.

  • Speaker: Steve Shkoller, University of California Davis
  • Title: The Geometry of Maximal Development of the Euler equations.
  • Abstract: I will describe a new geometric framework, new variables, and new uniform estimates for the maximal development of Cauchy data, which gives a complete characterization of the shock formation process for the multi-dimensional Euler equations of gas dynamics. This is joint work with Vlad Vicol.
  • Zoom link: https://vanderbilt.zoom.us/j/91288864129

Date: Friday, April 21st, 2023.

  • Speaker: Noah Lee, Princeton University 
  • Title: A Mathematical Study of Electroconvection
  • Abstract: We study electroconvective models mathematically described by the Nernst-Planck-Navier-Stokes (NPNS) or Nernst-Planck-Stokes (NPS) systems. These nonlocal, semilinear parabolic systems model the time evolution of ionic concentrations in a fluid in the presence of boundaries and an applied electrical potential on the boundaries. Ions diffuse under their own concentration gradients, are convected by the fluid, and are transported by the underlying electrical field. In turn, the electrical field is determined nonlocally by the distribution of ions and the applied electrical potential on the boundaries; the fluid is also forced by the electrical field.

    We consider these systems on three dimensional bounded domains, imposed with various equilibrium and nonequilibrium boundary conditions and address four main questions: 1) global existence of strong (smooth) solutions 2) existence, regularity, and boundedness of steady state solutions 3) long time dynamics of solutions, and 4) electroneutrality in the singular limit of zero Debye length ϵ → 0.

    One of the main features is the contrast of results between equilibrium and nonequilibrium boundary conditions. A primary difference between equilibrium and nonequilibrium boundary conditions is the existence and absence, respectively, of a natural dissipative structure for the corresponding NPNS/NPS system. In the case of equilibrium boundary conditions, the dissipative structure gives a natural starting point for further study of the dynamics of solutions and also gives a precise description of the asymptotic behavior of solutions in the limit of time t → ∞. On the other hand, for nonequilibrium boundary conditions, the lack of a dissipative structure manifests itself physically through more complex fluid patterns, which have been the subject of many experimental and numerical research efforts. In this thesis, we study this complex behavior from a rigorous mathematical viewpoint, considering both time independent and time dependent solutions, and in the latter case, considering their long time and long-time averaged behavior.