Category Archives: Uncategorized

Zhenhao Li (MIT)

The APDE seminar on Monday, 11/4, will be given by Zhenhao Li (MIT) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Quantum–classical correspondence past Ehrenfest time

Abstract: The evolution of a quantum system coupled to an external environment can be described by the Lindblad master equation. We consider such systems described by an at most quadratically growing classical Hamiltonian and self-adjoint jump operators (satisfying certain growth and nondegeneracy conditions). The classical counterpart to the Lindblad equation is given by a corresponding Fokker–Planck equation. We show that the quantum evolution remains close to the quantization of the classical evolution in trace norm for much longer than Ehrenfest time. The time of correspondence and the trace norm comparison improves upon recent works by Galkowski–Zworski and Hern{\’a}ndez–Ranard–Riedel in the constant to strong coupling strength regime.

Semyon Dyatlov (MIT)

The APDE seminar on Monday, 10/28, will be given by Semyon Dyatlov (MIT) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Control of eigenfunctions in higher dimensions

Abstract: Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. In previous work with Jin and Nonnenmacher we showed that for Laplacian eigenfunctions on negatively curved surfaces, semiclassical measures have full support. This was restricted to dimension 2 because the key new ingredient, the fractal uncertainty principle (proved by Bourgain and the speaker), was only known for subsets of the real line.

I will present several recent results on the support of semiclassical measures in higher dimensions, both on manifolds and in the toy model of quantum cat maps, contained in joint work with Jézéquel, joint work with Athreya and Miller, and work in progress by Kim. Some of these use the higher dimensional fractal uncertainty principle recently proved by Cohen. Others rely on separating the stable/unstable directions into fast and slow directions, and only applying the fractal uncertainty principle in the fast directions.

Gabriele Benomio (GSSI)

The APDE seminar on Monday, 10/14, will be given by Gabriele Benomio (GSSI) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: A new gauge for gravitational perturbations of Kerr spacetimes

Abstract: I will present a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range. Central to the framework is a new formulation of nonlinear gravitational perturbations of Kerr in a geometric gauge tailored to the outgoing principal null geodesics of Kerr. The main features of the framework will be illustrated in the context of the linearised theory, which serves as a fundamental building block in nonlinear applications.

Andrej Zlatos (UCSD)

The APDE seminar on Monday, 10/7, will be given by Andrej Zlatos (UCSD) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

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 develop a local well-posedness theory for this model in the stable regime (lighter fluid above the heavier one), which includes considerably more general fluid interface geometries than even existing whole plane results and allows the interface to touch the bottom.  The latter applies to the important scenario of the heavier fluid invading a region occupied by the lighter fluid along the impermeable layer.  We also show that finite time singularities do arise in this setting, including from arbitrarily small smooth initial data, by obtaining maximum principles for the height, slope, and potential energy of the fluid interface.

Koji Ohkitani (RIMS, Kyoto University)

The APDE seminar on Monday, 9/30, will be given by Koji Ohkitani (RIMS, Kyoto University) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Numerical determination of the self-similar profile
for the 3D Navier-Stokes equations and its applications

Abstract: We present the forward self-similar profile for the 3D Navier-Stokes
equations, representing the late stage of decaying Navier-Stokes flows.
The existence of such a profile has been known, but its precise functional
form has not been determined numerically, let alone mathematically.

Here we determine the profile for the first time using numerical methods.
This has been achieved by a combination of two things; a numerical method
of solving the Navier-Stokes equations in the whole space and the explicit
form of the linearised solution. Taking the initial data from the
linearised solution, we solve the fully-nonlinear Navier-Stokes equations
to observe its convergence to a steady solution in the dynamically scaled
space. We have confirmed that the nonlinear correction  is small,
consistent with the previous perturbative analysis. Applications of the
self-similar profile are briefly discussed.

Anuj Kumar (UC Berkeley)

The APDE seminar on Monday, 9/23, will be given by Anuj Kumar (UC Berkeley) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Nonuniqueness of solutions to the Euler equations with integrable vorticity

Abstract: Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. DiPerna and Majda proved the existence of weak solutions with vorticity in L^p ( p > 1).  A celebrated open question is whether the uniqueness result can be generalized to solutions with L^p vorticity. In this talk, we resolve this question in negative for some p > 1. To prove nonuniqueness, we devise a new convex integration scheme that employs non-periodic, spatially-anisotropic perturbations, an idea that was inspired by our recent work on the transport equation. To construct the perturbation, we introduce a new family of building blocks based on the Lamb-Chaplygin dipole. This is a joint work with Elia Bruè and Maria Colombo.

Warren Li (Princeton University)

The APDE seminar on Monday, 9/16, will be given by Warren Li (Princeton University) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: BKL bounces outside homogeneity

Abstract: In the latter half of the 20th century, physicists Belinski, Khalatnikov and Lifshitz (BKL) proposed a general ansatz for solutions to the Einstein equations possessing a (spacelike) singularity. They suggest that, near the singularity, the evolution of the spacetime geometry at different spatial points decouples and is well-approximated by a system of autonomous nonlinear ODEs, and further that general orbits of these ODEs resemble a (chaotic) cascade of heteroclinic orbits called “BKL bounces”. In this talk, we present recent work verifying the validity of BKL’s heuristics in a large class of symmetric, but spatially inhomogeneous, spacetimes which exhibit (up to one) BKL bounce on causal curves reaching the singularity. In particular, we prove AVTD behavior (i.e. decoupling) even in the presence of inhomogeneous BKL bounces. The proof uses nonlinear ODE analysis coupled to hyperbolic energy estimates, and one hopes our methods may be applied more generally.

Robert Schippa (UC Berkeley)

The first APDE seminar of this semester on Monday, 9/9, will be given by Robert Schippa (UC Berkeley) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Quantified decoupling estimates and applications

Abstract: In 2004 Bourgain proved a qualitative trilinear moment inequality for solutions to the Schrödinger equation on the circle and raised the question for quantitative estimates. Here we show quantitative estimates. The proof combines decoupling iterations with semi-classical Strichartz estimates. Related arguments allow us to extend Bourgain’s $L^2$-well-posedness result for the periodic KP-II equation to initial data with negative Sobolev regularity. One key ingredient are $L^4$-Strichartz estimates, which follow from a novel decoupling inequality due to Guth-Maldague-Oh. The latter part of the talk is based on joint work with Sebastian Herr and Nikolay Tzvetkov.

Hyunju Kwon (ETH Zürich)

The last APDE seminar of this semester on Monday, 5/6, will be given by Hyunju Kwon (ETH Zürich) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Strong Onsager conjecture

Abstract: Smooth solutions to the incompressible 3D Euler equations conserve kinetic energy in every local region of a periodic spatial domain. In particular, the total kinetic energy remains conserved. When the regularity of an Euler flow falls below a certain threshold, a violation of total kinetic energy conservation has been predicted due to anomalous dissipation in turbulence, leading to Onsager’s theorem. Subsequently, the $L^3$-based strong Onsager conjecture has been proposed to reflect the intermittent nature of turbulence and the local evolution of kinetic energy. This conjecture states the existence of Euler flows with regularity below the threshold of $B^{1/3}_{3,\infty}$ which not only dissipate total kinetic energy but also exhibit intermittency and satisfy the local energy inequality. In this talk, I will discuss the resolution of this conjecture based on recent collaboration with Matthew Novack and Vikram Giri.

Maxime Van de Moortel (Rutgers University)

The special APDE seminar on Thursday, 5/2, will be given by Maxime Van de Moortel (Rutgers University) in-person in Evans 748, and will also be broadcasted online via Zoom from 2:10pm to 3:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Polynomial decay in time for the Klein-Gordon equation on a Schwarzschild black hole

Abstract: It is expected that the Klein-Gordon equation on a Schwarzschild black hole behaves very differently from the wave equation at late-time, due to the presence of stable (timelike) trapping. We present our recent work demonstrating that despite the presence of stable timelike trapping on the Schwarzschild black hole, solutions to the Klein-Gordon equation with strongly localized initial data nevertheless decay polynomially in time. We will also explain how the proof uses, at a crucial step, results from analytic number theory related to the Riemann zeta function.
Joint works with Federico Pasqualotto and Yakov Shlapentokh-Rothman.