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 (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

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 (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

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 (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

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.

The APDE seminar on Monday, 4/15, will be given by Tarek Elgindi (Duke) 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 (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: Twisting in Hamiltonian flows and perfect fluids

Abstract: We will discuss a recent result joint with In-Jee Jeong and Theo Drivas. We prove that twisting in Hamiltonian flows on annular domains, which can be quantified by the differential winding of particles around the center of the domain, is stable to general perturbations. In fact, we prove the all-time stability of the lifted dynamics in an L2 sense (though single particle paths are generically unstable). These stability facts are used to establish several results related to the long-time behavior of inviscid fluid flows.

The APDE seminar on Monday, 4/8, will be given by Jens Wittsten (University of Borås) 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 (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: Semiclassical quantization conditions for strained moiré lattices.

Abstract: When mechanical strain is applied to bilayer graphene in a certain way, an essentially one-dimensional moiré pattern can be seen. I will discuss a model for such systems and explain that it has approximately flat bands when the strain is very weak. The approximately flat bands correspond to approximate eigenvalues of infinite multiplicity, and they are obtained by generalizing the Bohr-Sommerfeld quantization condition for scalar symbols at a potential well to matrix-valued symbols with eigenvalues that coalesce precisely at the bottom of the well. The talk is based on joint work with Simon Becker.