Columbia Analysis Seminar

Here are our upcoming seminars:

• September 15, 2023: Stefano Vita (University of Turin), Boundary Harnack principle on nodal domains

  Given a uniformly elliptic equation in divergence form, let us consider two solutions $u,v$ which share their zero sets $Z(u)\subseteq Z(v)$. Then, regularity features of the ratio $w=v/u$ across the nodal set of $u$ is equivalent to Schauder estimates for continuous solutions of some elliptic equations having coefficients which degenerate as $u^2$ on $Z(u)$.
  
  

• September 22, 2023 from 10:30am to 11:30am (NOTICE THE TIME CHANGE): John Anderson (Stanford University), Formation of shocks for the Einstein-Euler system

  In this talk, I hope to describe elements of proving stable shock formation for the Einstein-Euler system in the setting of potential flow. This involves proving that the fluid variables blow up in a specific way while the gravitational metric remains comparatively smooth. I'll first describe where this fits into the study of shocks, and why it is appropriate to call this singularity formation result a shock formation result. Then, I will go through some of the most important parts of Christodoulou's landmark proof of shock formation for potential flow on a fixed background, as well as followup breakthrough works by Luk-Speck allowing for vorticity. This will show the main difficulty present in proving that the gravitational metric remains comparatively smooth in the case of Einstien-Euler. It essentially arises from the fact that the speed of sound is less than the speed of light. In the remaining time, I will try to give the main idea in overcoming this difficulty. This is work in progress with Jonathan Luk.
  
  

• September 29, 2023: Chen-Chih Lai (Columbia University), Thermal effects in bubble oscillations

  We study the thermal decay of bubble oscillation in an incompressible liquid. In this talk, we consider two models, both systems of nonlinear PDEs with a moving boundary: the complete mathematical formulation (full model) and an approximate model proposed by A. Prosperetti in [J. Fluid Mech. 1991]. These two models share a one-parameter manifold of spherical equilibria, parametrized by the bubble mass. Within the approximate model, we prove that the manifold of spherical equilibria is an attracting centre manifold against small spherically symmetric perturbations and that solutions approach this manifold at an exponential rate as time advances. We also examine the dynamics of the bubble-fluid system subject to a small-amplitude, time-periodic external sound field. We prove that this periodically forced system admits a unique time-periodic solution that is nonlinearly and exponentially asymptotically stable against small spherically symmetric perturbations. Finally, we report on results regarding the characterization of all equilibria within each model. For the approximate system, we prove that all equilibrium bubbles are spherically symmetric through an application of Alexandrov’s theorem on closed constant-mean-curvature surfaces. Furthermore, the aforementioned family of spherical equilibria encompasses all the spherical equilibria of the approximate system. However, within the full model, this family is embedded in a larger family of spherically symmetric solutions. If time permits, I will discuss a work in progress on asymmetric dynamics of these models and future directions. This talk is based on joint work with Michael I. Weinstein (arXiv:2207.04079, arXiv:2305.03569, and work in progress).
  
  

• October 6, 2023: Ravi Shankar (Princeton University), TBA

  TBA
  
  

• October 27, 2023: Tristan Ozuch (MIT), TBA

  TBA
  
  

• November 3, 2023: Ryan Unger (Princeton University), TBA

  TBA
  
  

• November 10, 2023: Donatella Danielli (Arizona State University), TBA

  TBA
  
  

• November 17, 2023: Demetre Kazaras (Michigan State University), TBA

  TBA
  
  

• December 1, 2023: Allen Fang (Munster University), TBA

  TBA
  
  

• December 8, 2023: Lili He (Princeton University), TBA

  TBA
  
  

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The organizers,

Daniela De Silvia, Elena Giorgi