Speaker: Noa Zilberman Princeton University
Title: Quantum effects inside black holes
Abstract: Astrophysical black holes are known to be rotating. Within classical General Relativity, the simplest spacetime solution (the Kerr solution) describing a rotating black hole reveals a traversable passage through an inner horizon - which in turn may lead to another external universe. But does this remain the case when taking quantum effects into account? Answering this question, along others, requires one to understand the manner in which quantum energy fluxes affect the internal geometry of a black hole. It has been widely anticipated, yet inconclusive (till this work), that such effects would diverge at the inner horizon of a spinning black hole. This divergence, if indeed takes place, may drastically affect the internal black hole geometry, potentially preventing the inner horizon traversability. Clarifying this issue requires the computation of the quantum energy fluxes in black hole interiors. However, this has been a serious challenge for decades. Using a combination of old and new methods, we have managed to compute the quantum energy fluxes at the inner horizon of a spinning black hole, in a vacuum state corresponding to an evaporating black hole. We found that these fluxes are either positive or negative, depending on the black hole spin (and polar angle). The sign of these fluxes may be crucial to the nature of their backreaction on the geometry (as should be dictated by the semiclassical Einstein equation). In this seminar, we shall briefly describe the basic framework of semiclassical general relativity and the renormalization procedure, and then present our novel results for the quantum fluxes at the inner horizon of a rotating black hole, briefly mentioning possible implications for the inner horizon traversability.
Speaker: Jia Shi MIT
Title: Non-radial implosion for compressible Euler, Navier-Stokes and defocusing NLS in $\mathbb{T}^d$ and $\mathbb{R}^d
Abstract: We will discuss the smooth, non-radial solutions of the compressible Euler, Navier-Stokes equation and defocusing nonlinear Schrodinger equation that develop an imploding finite time singularity. The construction is motivated by the radial imploding solutions from Merle-Raphael-Rodnianski-Szeftel, and Cao-Labora-Buckmaster-Gomez-Serrano but is flexible enough to handle both periodic and non-radial initial data. This is a joint work with Gonzalo Cao-Labora, Javier Gomez-Serrano, and Gigliola Staffilani.
Speaker: Gong Chen GeorgiaTech
Title: Recovery of the nonlinearity from the modified scattering map
Abstract: We consider the problem of recovering the nonlinearity in a nonlinear Schr\”odinger equation from scattering data, a problem for which there is a relatively large literature. We consider a new situation in which the equation does not admit standard scattering, but instead features the modified scattering behavior with logarithmic phase correction. We prove that even in this case, the modified scattering data suffices to determine the unknown nonlinearity. This is a joint work with J. Murphy (Oregon)
Speaker: Sohrab Shahshahani University of Massachusetts, Amherst
Title: Codimension one stability of the catenoid for the hyperbolic vanishing mean curvature equation
Abstract: The catenoid, which is a minimal surface, can be viewed as a stationary solution of the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. The main result discussed in this talk is the nonlinear asymptotic stability, modulo suitable translation and Lorentz boost (i.e., modulation), of the n-dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for n = 5 and n = 3. The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. In a broader context, these works fit in the long tradition of studies of soliton stability problems. From this viewpoint, they outline a systematic approach for studying soliton stability problems for quasilinear wave equations. This talk is based on joint works with Jonas Luhrmann and Sung-Jin Oh.
Speaker: Yan Guo Brown University
Title: Stability of Contact Lines
Abstract: Contact lines (e.g. coffee meets with the coffee cup, boundary of a droplet on table) occurs naturally when a free fluid surface, a capillary surface, meets with a fixed solid boundary. Despite its importance in fluid theory and applications, the dynamical law for contact line has been only studied in relative recent history. We review recent work to establish well-posedness of classical fluid models for describing contact line dynamics.
Speaker: Xuwen Chen University of Rochester
Title: Well/Ill-posedness separation of the Boltzmann equation with cut-off
Abstract: We report the finding of the sharp separation of well/ill-posedness for the Boltzmann equation with cut-off using dispersive PDE techniques. The separation is unexpectedly 1/2-derivative above scaling and the ill-posedness is represented by forward in time norm deflation.
Speaker: Ioakeim Ampatzoglou Baruch College, CUNY
Title: Scattering theory for the inhomogeneous kinetic wave equation
Abstract: We will talk about the construction of global strong dispersive solutions to the space inhomogeneous kinetic wave equation (KWE) which propagate $L^1_{xv}$ -- moments and conserve mass, momentum and energy. We prove that they scatter, and that the wave operators mapping the initial data to the scattering states are 1-1, onto and continuous in a suitable topology. Our proof is carried out entirely in physical space, and combines dispersive estimates for the free transport with new trilinear bounds for the gain and loss operators of the KWE on weighted Lebesgue spaces. A fundamental tool in obtaining these bounds is a novel collisional averaging estimate. Finally we show that the nonlinear evolution preserves positivity forward in time. For this, we use the Kaniel-Shinbrot iteration scheme, properly initialized to ensure the successive approximations are dispersive.
Speaker: Andrej Zlatos University of California, San Diego
Title: Stable regime singularity for the Muskat problem
Abstract: The Muskat problem on the half-plane models motion of an interface between two fluids of distinct densities in a porous medium that sits atop an impermeable layer, such as oil and water in an aquifer above bedrock. We show that unlike on the whole plane, finite time singularities do arise in the stable regime (lighter fluid above the heavier one) in this setting, including from arbitrarily small smooth initial data. We achieve this by developing a local well-posedness theory for this model as well as obtaining maximum principles for the height, slope, and potential energy of the fluid interface. The former allows the interface to touch the bottom, which applies to the important scenario of the heavier fluid invading a region occupied by the lighter fluid along the impermeable layer, and includes considerably more general fluid interface geometries than even previous whole plane results.
Speaker: Justin Holmer Brown University
Title: A derivation of the Boltzmann equation from quantum many-body dynamics
Abstract: We start by introducing a statistical model for the initial data of an N-body Schrodinger equation, meant to represent a scaled version of an N-particle quantum system with unit-order velocities and interparticle separations. The statistical model yields the expected functional form and scale of the corresponding BBGKY densities. This motivates a general a priori assumption on the Sobolev space norms of the BBGKY densities, which includes quasi-free states. Under this assumption, we prove that the Wigner transformed densities converge to the Boltzmann hierarchy with quadratic collision kernel and quantum scattering cross section. The proof of convergence uses a framework previously applied to the derivation of Bose Einstein condensate from an N-body model, and involves exploiting uniform bounds to obtain compactness and weak convergence. The remaining step is to prove the uniqueness of limits, which is performed using the Hewitt-Savage theorem and an extension of the Klainerman-Machedon board game. Our derivation is optimal with respect to regularity considerations. This is joint work with Xuwen Chen, University of Rochester.
Speaker: John Stalker Trinity College Dublin
Title: Prescribing initial data for the Einstein equations at a singularity
Abstract: Initial point singularities are of interest in general relativity because considerable evidence suggests that our universe had one. Prescribing initial data for such solutions presents some obvious problems though. Work over the last few decades has explored how to do this for a variety of reasonable matter models though, including polytropic fluids, electromagnetic radiation, and kinetic theory, with or without collisions. This talk will mostly be an overview but I will briefly discuss some recent work with Ho Lee, Ernesto Nungesser and Paul Tod.