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.
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.
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.
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.
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.
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.
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.
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.
Speaker: Gavin Stewart Rutgers University
Title: Scattering and localized states for nonlinear Schrodinger equations
Abstract: In this talk, I will discuss some results on the long time behavior of nonlinear Schrodinger equations with potentials. First, I will show that solutions which are initially localized and nonradiative can only spread slowly in space. Then, I will focus on the mass supercritical equation, and show that solutions decompose into a free wave and a slowly spreading remainder. A key ingredient in these arguments is the use of Morawetz and interaction Morawetz estimates localized to an exterior region in space-time. This is joint work with Avy Soffer.
Speaker: Jaydeep Singh Princeton University
Title: Regimes of stability for self-similar naked singularities
Abstract: The family of k-self-similar naked singularities, first constructed rigorously by Christodoulou, offers a concrete setting for exploring the instability mechanisms underlying weak cosmic censorship. In this talk we introduce these spacetimes, as well as the classical argument for a blue-shift instability at low regularity. We then consider classes of perturbations supported in the exterior and interior regions, building a picture of how the blue-shift behaves as a function of the support and regularity of initial data considered.
Speaker: Joonhyun La Princeton University
Title: Null shell solutions: stability and instability
Abstract: In this talk, we study initial value problem for the Einstein equation with null matter fields, motivated by null shell solutions of Einstein equation. In particular, we show that null shell solutions can be constructed as limits of spacetimes with null matter fields. We also study the stability of these solutions in Sobolev space: we prove that solutions with one family of null matter field are stable, while the interaction of two families of null matter fields can give rise to an instability. The talk is based on a joint work in progress with Jonathan Luk (Stanford).
Speaker: John Anderson Stanford University
Title: Formation of shocks for the Einstein-Euler system
Abstract: In this talk, I hope to describe elements of proving a certain stable singularity formation result for the Einstein-Euler system, which is the topic of work in progress with Jonathan Luk. I'll first describe where this fits into the big picture of the study of multidimensional shocks, and why it is appropriate to call this a shock formation result. Then, I will try to describe some of the main ideas that go into proving shock formation, and the main difficulty in the case of Einstein-Euler. In a nutshell, the difficulty arises from the fact that the speed of sound is less than the speed of light. In the remaining time, I will describe how this is related to shocks for other hyperbolic PDEs arising in continuum mechanics.
Speaker: Toan Nguyen PennState University
Title: Survival threshold
Abstract: The talk is to identify the survival threshold of wave numbers which completely characterizes the large time dynamics of interacting charged or quantum particles modelled by their respective meanfield models such as Vlasov and Hartree equations near translation-invariant steady states. This consists of pure plasma oscillations, phase mixing, and Landau damping due to resonances.
Speaker: André Guerra ETH Zurich
Title: Homogenization in General Relativity
Abstract: Given their highly nonlinear nature, Einstein's vacuum equations are not closed under weak convergence and hence sequences of weakly-convergent solutions may generate a non-trivial energy momentum tensor in the limit. In 1989 Burnett conjectured that, for a sequence of vacuum solutions which oscillates with high frequency, this limit is characterized by the Einstein-massless Vlasov model: in particular, starting from vacuum, matter is generated through homogenization. In this talk we will present a proof of this conjecture, under appropriate gauge and symmetry assumptions. Based on joint work with Rita Teixeira da Costa (Princeton University).
Speaker: Avy Soffer Rutgers University
Title: Scattering Theory of Linear and Nonlinear Waves: A Unified New Paradigm
Abstract: I will present a new approach to Mathematical Scattering of multichannel Dispersive and Hyperbolic Equations. In this approach we identify the targe time behavior of such equations, both linear and non-linear, for general (large) data, and interactions terms which can be space-time dependent. In particular, for the NLS equations with spherically symmetric data and Interaction terms, we prove that all global solutions in H^1 converge to a smooth and localized function plus a free wave, in 5 or more dimensions. Similar result holds for 3,4 dimensions, though the argument proving localization is different. We also show similar results in any dimension for localized type of interactions, provided they decay fast enough. We show breakdown of the standard Asymptotic Completeness conjecture if the interaction is time dependent and decays like r^{-2} at infinity. Many of these results extend to the non-radial case, for NLS, NLKG and Bi-harmonic NLS in three or more dimensions. Furthermore, we prove Local-Decay Estimates for Time dependent potentials in 5 or more dimensions. Finally, we apply this approach to N-body scattering, and prove AC for three quasi-particle scattering. This is based on joint works with Baoping Liu and Xiaoxu Wu.
Speaker: Ebru Toprak Yale University
Title: $L^1 \to L^\infty$ dispersive estimates for Coulomb waves
Abstract: In this talk, I will present our recent result on the spherically symmetric Coulomb waves. We study the evolution operator of $H=-\Delta+q |x|^{-1}$, with $q>0$. By means of a distorted Fourier transform adapted to $H$, we compute the evolution kernel explicitly. A detailed analysis of this kernel shows that $e^{i t H}$ obeys an $L^1 \to L^\infty$ dispersive estimate with the natural decay rate $t^{-\f32}$. This is a joint work with Adam Black, Bruno Vergara, Jiahua Zhou.
Speaker: Sanchit Chaturvedi New York University/ Simons Institute
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).
Speaker: Jared Speck Vanderbilt University
Title: The structure of the maximal development for shock-forming 3D compressible Euler solution
Abstract: I will discuss my ongoing series of works with L. Abbrescia on 3D compressible Euler flow. In these works, we derive the complete structure of a localized portion of the maximal development for open sets of initial data without symmetry, irrotationality, or isentropicity assumptions. In particular, we describe the full structure of a localized portion of the singular set, where the solution’s gradient blows up in a shock singularity, as well as the emergence of a localized piece of a Cauchy horizon from the singularity. Our work builds on Christodoulou’s breakthrough monographs on irrotational solutions and my prior works with J. Luk, which revealed an implicit portion of the singular set. The key new ingredients are rough foliations of spacetime adapted to the shape of the boundaryand a geo-analytic framework that yields suitable estimates on the foliations. Finally, I will discuss some of the many open problems in the field.
Speaker: Alberto Bressan PennState University
Title: Uniqueness and error estimates for hyperbolic conservation laws.
Abstract: In this talk I shall review some old and new results about uniqueness of solutions to hyperbolic conservation laws. In particular: for any n x n strictly hyperbolic system, any weak solution which takes values inside the domain of the semigroup of vanishing viscosity limits, and whose shocks satisfy the Liu admissibility conditions, actually coincides with a semigroup trajectory. Implications of his result toward a posteriori error estimates will be discussed.