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.

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.

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.

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.

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.

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.

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.

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.

The APDE seminar on Monday, 4/29, will be given by In-Jee Jeong (Seoul National 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: Opportunities for the SQG equation

Abstract:We review various attempts in the proof of singularity formation and their limitations for the inviscid surface quasi-geostrophic (SQG) equation. The key difficulty can be summarized as (unexpected) cancellation and regularizing structure of the nonlinearity. Then we discuss remaining opportunities for the proof of singularity formation, in the class of relatively low regularity data.

The APDE seminar on Monday, 4/22, will be given by Dongxiao Yu (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: Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms

Abstract: I will present a joint work with Herbert Koch on the asymptotic stability of the sine-Gordon kinks under small perturbations in weighted Sobolev norms. Our main tool is the Bäcklund transform which reduces the study of the asymptotic stability of the kinks to the study of the asymptotic decay of solutions near zero. I will also compare our work with some previous work on the asymptotic stability of the sine-Gordon kinks.