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: Generalized Price’s law on fractional-order asymptotically flat stationary spacetimes
  • Abstract: Price’s law is a conjecture from the 70’s by physicist R. Price predicting that waves on the Schwarzschild spacetime decay pointwise at a rate of 1/t^3. The Schwarzschild geometry, which describes spacetime in the presence of a stationary black hole, is a long range perturbation of the flat Minkowski metric. On the other hand, we know that waves decay infinitely fast on Minkowski space as can be seen by sharp Huygens’ principle. In the current work we study decay rates of waves on stationary spacetimes which are short range perturbations of Minkowski space. In particular, the geometries considered have a prescribed rate at which they tend toward flat as the size of the spatial variables tends toward infinity. We describe the pointwise decay rate of waves in terms of this prescribed rate toward flatness. We are specifically interested in the case where the prescribed rate takes on non-integer values as the integer case was considered in previous work by M. The background geometries are allowed to exhibit weak trapping. This work shows it is the far away behavior of an asymptotically flat stationary spacetime which dictates the local decay rate of waves. This is joint work with Jared Wunsch.

Date: Friday, Nov. 5th, 2021.

  • Speaker: Junyan Zhang, Johns Hopkins University.
  • Title: Anisotropic regularity of the free-boundary problem in ideal compressible MHD
  • Abstract: We consider the free-boundary ideal compressible MHD system under the Rayleigh-Taylor sign condition. The local well-posedness was recently proved by Trakhinin and Wang by using Nash-Moser iteration. We prove the a priori estimate without loss of regularity in the anisotropic Sobolev space. Our proof is based on the combination of the “modified” Alinhac good unknown method, the full utilization of the structure of MHD system and the anisotropy of the function space. This is joint work with Professor Hans Lindblad.
  • Zoom Link: https://vanderbilt.zoom.us/j/97469717946

Date: Friday, Nov. 12th, 2021.

  • Speaker: Federico Pasqualotto, Duke University.
  • Title: Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation
  • Abstract: In this talk, I will discuss a construction of “shock forming” solutions to a class of dispersive and dissipative perturbations of the Burgers equation. This class includes the fractional KdV equation with dispersive term of order α in [0,1), the Whitham equation arising in water waves, and the fractal Burgers equation with dissipation term of order β in [0,1).Our result seems to be the first construction of gradient blow-up for fractional KdV in the range α in [2/3,1). We construct blow-up solutions by a self-similar approach, treating the dispersive term as perturbative.The blow-up is stable for α < 2/3. However, for α ≥ 2/3, the solution is constructed by perturbing an underlying unstable self-similar Burgers profile. The construction is carried out by means of a weighted L^2 approach, which may be of independent interest.This is joint work with Sung-Jin Oh.

Date: Friday, Dec. 3rd, 2021.

  • Speaker: Ákos Nagy, UC Santa Barbara.
  • Title: Keller–Segel equation on curved planes
  • Abstract: The Keller–Segel equations provide a mathematical model for chemotaxis, that is the organisms (typically bacteria) in the presence of a (chemical) substance. These equations have been intensively studied on R^n with its flat metric, and the most interesting and difficult case is the planar, n = 2. Less is known about solutions in the presence of nonzero curvature.
    In the talk, I will introduce the Keller–Segel equations in dimension 2, and then briefly recall a few relevant known facts about them. After that I will present my main results. First I prove sharp decay estimates for stationary solutions and prove that such a solution must have mass 8π. Some aspects of this result are novel already in the flat case. Furthermore, using a duality to the “hard” Kazdan–Warner equation on the round sphere, I prove that there are arbitrarily small perturbations of the flat metric on the plane that do not support a stationary solution to the Keller–Segel equations.
    I will also prove a curved version of the logarithmic Hardy–Littlewood–Sobolev inequality, which I use to prove a result that is complementary to the above ones, as it shows that the functional corresponding to the Keller–Segel equations is bounded from below only when the mass is 8π.
    Finally, if time permits, I will present a few results about the nonstationary case, that is a work in progress. In particular, I show long time existence for small masses for certain metrics.
    • This talk will take at a different time than our usual seminars: SC 1312, 3:30 pm.

Date: Wednesday, Dec. 8th, 2021.

  • Speaker: Maxime Van de Moortel, Princeton University.
  • Title: Black holes: the inside story.
  • Abstract:What does the interior of a black hole look like? Beyond the astrophysical motivation, it turns out that this question is at the heart of profound conjectures in General Relativity. One of them is the celebrated Penrose’s Strong Cosmic Censorship supporting the deterministic character of the classical theory of gravitation.
    I will present recent advances on this topic based on modern mathematical techniques and describe outstanding problems.
  • Zoom link: https://vanderbilt.zoom.us/j/96932716173
  • This talk will take at a different time and place than our usual seminars: Science and Engineering Building, ESB 044, 10:30am.