Hyperbolic & Dispersive PDE Seminar

Past talks (2023-2024)


Spring 2024

May 9, 2024, Thursday, 3:50-4:50pm, Hill 705

Speaker: Daniel Eceizabarrena University of Massachusetts Amherst

Title: Multifractality in the evolution of vortex filaments

Abstract: Vortex filaments that evolve according the binormal flow are expected to exhibit turbulent properties. Aiming to quantify this, I will discuss the multifractal properties of the family of functions $R_{x_0}(t) = \sum_{n=0} \frac{e^{2 \pi i (n^2 t+nx_0)}}{n^2}, x_0 ? [0, 1]$, that approximate the trajectories of regular polygonal vortex filaments. These functions are a generalization of the classical Riemann's non-differentiable function,which we recover when x0 = 0. I will highlight how the analysis seems to critically depend on x0, and I will discuss the important role played by Gauss sums, a restricted version of Diophantine approximation, the Duffin-Schaeffer theorem, and the mass transference principle. This talk is based on the article https://arxiv.org/abs/2309.08114 in collaboration with Valeria Banica (Sorbonne Universite), Andrea Nahmod (University of Massachusetts) and Luis Vega (BCAM, UPV/EHU).

April 18, 2024, Thursday, 3:50-4:50pm, Hill 705

Speaker: Michael Weinstein Columbia University

Title: Free boundary problems arising in the dynamics of a gas bubble in an unbounded liquid

Abstract: Consider a deforming gas bubble immersed in an unbounded liquid with surface tension. For an incompressible and inviscid liquid Rayleigh found (in the linearized approximation) that a spherically symmetric equilbrium bubble is neutrally stable; an infinitesimally small perturbation excites undamped oscillations in all spherical harmonic components of the perturbation. However, an actual fluid-bubble system comes with dissipation mechanisms: thermal diffusion, viscosity, and in the case where the liquid is compressible, acoustic radiation. We discuss recent results on the thermal decay of radial bubble oscillations in an incompressible liquid. We focus on the approximate (isobaric) model of A. Prosperetti [J. Fluid Mech. 1991]; see also Biro-Velazquez [SIMA 2000]. i) We show that if the liquid has non-zero viscosity and surface tension, then all equilibrium bubbles are spherically symmetric by an application of Alexandrov’s theorem on closed constant-mean-curvature surfaces. ii) The model exhibits a one-parameter manifold of spherically symmetric equilibria (steady states), which is parametrized by the bubble mass (encompassing all regular spherical equilibria). We prove that the manifold of spherical equilibria is an attracting centre manifold relative to small spherically symmetric perturbations, and that solutions approach this manifold at an exponential rate as time advances. iii) We also study 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. For the general asymmetric dynamics about a spherical bubble equilibria, we expect the deforming bubble to eventually relax to a sphere. Iwill discuss work in progress on asymmetric dynamics of these models and future directions.

April 12, 2024, Friday, 3:30-4:30pm, Hill 705 (Colloquium)

Speaker: Jonathan Luk Stanford University

Title: Late time tails of linear and nonlinear waves

Abstract: I will present a recent joint work with Sung-Jin Oh (Berkeley), where we develop a general method for understanding the precise late time asymptotic behavior of solutions to linear and nonlinear wave equations in odd spatial dimensions. In particular, we prove that in the presence of a nonlinearity and/or a dynamical background, the late time tails are in general different from the better understood case of linear equations on stationary backgrounds. I will explain how the late time tails are related to the problem of the singularity structure in the interior of generic dynamical vacuum black holes in general relativity.

March 28, 2024, Thursday, 3:50-4:50pm, Hill 705

Speaker: Tristan Leger Princeton University

Title: On the cubic NLS equation with a trapping potential

Abstract: In this talk I will start by motivating the study of the cubic NLS equation with harmonic potential: several physical situations where it arises will be presented, putting special emphasis on turbulent phenomena. I will then present (old and new) results from the mathematical theory developed to study this equation.

March 19, 2024, Tuesday, 1:40-2:40pm (Special day/time), Hill 705. Joint with Nonlinear Analysis Seminar.

Speaker: Lili He Princeton 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.

February 27, 2024, Tuesday, 1:40-2:40pm (Special day/time), Hill 705. Joint with Nonlinear Analysis Seminar.

Speaker: Hans Ringstrom KTH

Title: A quiescent regime for big bang formation

Abstract: Recently, many results concerning stable big bang formation have appeared. Most of the results concern stability of spatially homogeneous and isotropic solutions. However, a recent result of Fournodavlos, Rodnianski and Speck (FRS) covers the full regime in which stability is to be expected. On the other hand, it is restricted to the stability of spatially homogeneous and spatially flat solutions. In this talk, I will present a new result (joint work with Hans Oude Groeniger and Oliver Petersen) in which we identify a general condition on initial data ensuring big bang formation. The solutions need, in this case, not be close to symmetric background solutions. Moreover, the result reproduces previous results in the Einstein-scalar field and Einstein-vacuum settings. Finally, the result is in the Einstein-non-linear scalar field setting, and therefore yields future and past global non-linear stability of large classes of spatially locally homogeneous solutions.

February 22, 2024, Thursday, 3:50-4:50pm, Hill 705

Speaker: Sam Collingbourne Columbia University

Title: Uniform Boundedness for Linearised Gravity on Schwarzschild from the Canonical Energy

Abstract: In this talk, I will discuss a robust method for producing a conservation law for linearised gravity on a stationary spacetime: the canonical energy. Whilst its coercivity properties are obscure, I will argue that it can be used to obtain a uniform boundedness result for the gauge invariant Teukolsky variables on Schwarzschild. Remarkably, this does not rely on the (decoupled) Teukolsky equation itself or the transformation theory associated to it. Time permitting I will discuss how this makes it a good candidate method for exploring higher dimensional spacetimes where the decoupling and transformation theory fails. This work is joint with Gustav Holzegel.

February 16, 2024, Friday, 2:00-3:00pm, Hill 425 (Special time/day and location)

Speaker: Marta Lewicka University of Pittsburgh

Title: The Monge-Ampere system and the isometric immersion system: convex integration in arbitrary dimension and codimension

Abstract: The Monge-Ampere equation \det\nabla^2 v =f posed on a d=2 dimensional domain \omega and in which we are seeking a scalar (i.e. dimension k=1) field v, has a natural weak formulation that appears as the constraint condition in the \Gamma-limit of the dimensionally reduced non-Euclidean elastic energies. This formulation reads: \frac{1}{2}\nabla v\otimes \nabla v) + \sym\nabla w= - (\curl \curl)^{-1}f and it allows, via the Nash-Kuiper scheme of convex integration, for constructing multiple solutions that are dense in C^0(\omega), at the regularity C^{1,\alpha} for any \alpha<1/3, no matter the sign of the right hand side function f. Does a similar result hold in higher dimensions d>2 and codimensions k>1? Indeed it does, but one has to replace the Monge-Ampere equation by the Monge-Ampere system, by altering \curl \curl to the corresponding operator that arises from the prescribed Riemann curvature problem, similarly to how the prescribed Gaussian curvature problem leads to the Monge-Ampere equation in 2d. Our main result is a proof of flexibility of the Monge-Ampere system at C^{1,\alpha} for \alpha<1/(1+d(d+1)/k). This finding extends our previous result where d=2, k=1, and stays in agreement with the known flexibility thresholds for the isometric immersion problem: the Conti-Delellis-Szekelyhidi result \alpha<1/(1+d(d+1)) when k=1, as well as the Kallen result where \alpha\to 1 as k\to\infty. For d=2, the flexibility exponent may be even improved to \alpha<1/(1+2/k), using the conformal invariance of 2d metrics to the flat metric. We will also discuss other possible improvements and parallel results and techniques valid for the isometric immersion system.

February 1, 2024, Thursday, 3:50-4:50pm, Hill 705

Speaker: Narek Hovsepyan Rutgers University

Title: On the lack of external response of a nonlinear medium in the second-harmonic generation process.

Abstract: Second Harmonic Generation is a process in which the input wave (e.g. laser beam) interacts with a nonlinear medium and generates a new wave, called the second harmonic, at double the frequency of the original input wave. We investigate whether there are situations in which the generated second harmonic wave does not scatter and is localized inside the medium, i.e., the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated inside the medium with non-standard boundary conditions. This is based on a joint work with F. Cakoni, M. Lassas and M. Vogelius.

January 25, 2024, Thursday, 3:50-4:50pm, Hill 705

Speaker: Stan Palasek IAS/Princeton University

Title: Non-uniqueness of weak solutions for forced fluid equations

Abstract: Convex integration has for a decade now been used to produce impressive examples of the flexibility of weak solutions to fluid equations. Still, for many models there remains a gap between the regularity of the construction and the known or conjectured regularity at which uniqueness holds for the Cauchy problem. In this talk we present a new alternating approach to convex integration for forced models. As a consequence, one obtains non-uniqueness results for forced 2D Euler, 3D Euler, and SQG, in some cases with higher regularity than possible with previous convex integration schemes. This is joint work with Aynur Bulut and Manh Khang Huynh.

Fall 2023