PDE Seminar Fall 2019

Fridays, 4:10pm, Stevenson Center 1307

Date: Friday, Sept 13, 2019.

  • Speaker: Zoe Wyatt, University of Edinburgh.
  • Title: Attractors of the Einstein-Klein-Gordon system.
  • Abstract: A key question in general relativity is whether solutions to the Einstein equations, viewed as an initial value problem, are stable to small perturbations of the initial data. For example, previous results have shown that the Milne spacetime, which represents an expanding universe emanating from a big bang singularity with a linear scale factor, is a stable solution to the Einstein equations. With such a slow expansion rate, particularly compared to related models with accelerated expansion (such as the exponentially expanding de Sitter spacetime modelling our universe), there are interesting questions one can ask about stability of this spacetime. Previous results have shown that the Milne model is a stable solution to the vacuum Einstein, Einstein-Klein-Gordon and Einstein-Vlasov systems. Motivated by techniques from the last result, I will present a new proof of the stability of the Milne model to the Einstein-Klein-Gordon system and compare our method to a recent result of J. Wang. This is joint work with David Fajman (Vienna).

Date: Friday, Sept 27, 2019.

  • Speaker: Vu Hoang, The University of Texas at San Antonio.
  • Title: The mathematical physics of the relativistic radiation reaction.
  • Abstract: In classical electrodynamics, the self-force of a charged particle on itself is undefined, leading to a major inconsistency in the equations of motion for point particles. The usual solution of the problem involves a renormalization procedure, which is mathematically not well defined. It appears that a successful solution of this problem requires a modification of the Maxwell-Lorentz equations. In this talk, I consider the field equations of higher-order electrodynamics as proposed by Bopp, Lande, Thomas and Podolsky in the 1940’s. This theory modifies the Maxwell Lagrangian by adding terms containing higher-order derivatives of the field tensor, leading to higher-order wave equations that couple charged particles to their radiation fields. Starting with a brief review of the subject’s history, I will focus on recent results by Kiessling-Tahvildar-Zadeh and by M. Radosz and myself, showing that a covariant equation of motion for point particles can be rigorously derived from the postulate of conservation of energy and momentum without any renormalization. I will also present a global existence result for the motion of a single particle under the influence of an external field and its radiation reaction. This is joint work with M. Radosz.

Date: Friday, Oct 18, 2019.

  • Speaker: Mihaela Ifrim, University of Wisconsin Madison.
  • Title: Well-posedness and dispersive decay of small data solutions to the Benjamin-Ono equation.
  • Abstract: This result represents a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation on the real line. While this problem is known to be both completely integrable and globally well-posed in L2, much less seems to be known concerning its long time dynamics. Here, we prove that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the L2 theory for the Benjamin-Ono equation and provide a simpler, self-contained approach.

Date: Friday, Nov 15, 2019.

  • Speaker: Yuanzhen Shao, University of Alabama.
  • Title: Large Time Behavior of the Fractional Porous Medium Equation on Riemannian Manifolds via Fractional Logarithmic Sobolev Inequalities.
  • Abstract: In 1995, Carlen and Loss proposed a novel approach to study the asymptotics the 2–D Navier–Stokes equation by using a Logarithmic Sobolev inequality. Later, this idea was adopted by Bonforte and Grillo to study the asymptotics of the Porous Medium Equation. In this talk, I will discuss how to derive various Sobolev type inequality involving fractional Laplacian. Then I will use these inequalities to study the smooth effect and asymptotic behavior of the Fractional Porous Medium Equation on complete Riemannian manifolds.

Date: Friday, Dec 6, 2019.

  • Speaker: Theodore Drivas, Princeton University.
  • Title: Anomalous dissipation for passive scalars.
  • Abstract: We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. We give an example of a rough divergence-free velocity field that explicitly exhibits anomalous dissipation for passive scalars. The mechanism for scalar dissipation is a built-in direct energy cascade in the synthetic velocity field. Connections to the Obukhov–Corrsin mono fractal theory of scalar turbulence and to inviscid mixing will be discussed. This is joint work with T. Elgindi, G. Iyer and I-J Jeong.