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.

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 () or Mengxuan Yang ().

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 () or Mengxuan Yang ().

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.

The APDE seminar on Monday, 4/1, will be given by Vera Mikyoung Hur 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 undular bores: rigorous analysis and validated numerics 

Abstract: I will discuss the ‘global’ nonlinear asymptotic stability of the traveling front solutions to the Korteweg-de Vries–Burgers equation, and other dispersive-dissipative perturbations of the Burgers equation. Earlier works made strong use of the monotonicity of the profile, for relatively weak dispersion effects. We exploit the modulation of the translation parameter, establishing a new stability criterion that does not require monotonicity. Instead, a certain Schrodinger operator in one dimension must have exactly one negative eigenvalue, so that a rank-one perturbation of the operator can be made positive definite. Counting the number of bound states of the Schrodinger equation, we find a sufficient condition in terms of the ’width’ of a front. We analytically verify that our stability criterion is met for an open set in the parameter regime including all monotone fronts. Our numerical experiments, revealing more stable fronts, suggest a computer-assisted proof. Joint with Blake Barker, Jared Bronski, and Zhao Yang.

The APDE seminar on Monday, 3/18, will be given by Zhuolin Li (SLMath/MSRI) 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: Degenerate variational problems under the constant rank condition

Abstract: Differential expressions involving non-elliptic operators emerge in various PDEs and variational principles that arise from materials science, fluids, differential geometry, etc. Despite their inherent degeneracy, such operators, under the constant rank condition, retain certain good properties of elliptic operators. In this talk, we will first give a short introduction to the study of vectorial problems in the calculus of variations, and then discuss quasi-convex variational problems involving constant rank operators. For clarity, exterior derivatives will be taken as a particular example for illustration. We will consider the existence, which can also be interpreted as a Sobolev-type regularity, as well as the corresponding partial regularity via an excess decay estimate strategy. This talk is based on an ongoing work with Bogdan Raiță.

The APDE seminar on Wednesday, 3/13, will be given by Jeremy Marzuola (UNC) in-person in Evans 748, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang (). Please note the special time and location of this talk.

Title:  Spectral minimal partitions, nodal deficiency and the Dirichlet-to-Neumann map

Abstract:  The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to compute, or even estimate. Here we compare two recently obtained formulas for the nodal deficiency, one in terms of an energy function on the space of equipartitions of the manifold, and the other in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We relate these two approaches by giving an explicit formula for the Hessian of the equipartition energy in terms of the Dirichlet-to-Neumann map. This allows us to compute Hessian eigenfunctions, and hence directions of steepest descent, for the equipartition energy in terms of the corresponding Dirichlet-to-Neumann eigenfunctions. Our results do not assume bipartiteness, and hence are relevant to the study of spectral minimal partitions.  This is joint work with Greg Berkolaiko, Yaiza Canzani and Graham Cox.

The APDE seminar on Monday, 3/11, will be given by Hezekiah Grayer (Princeton) 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:  On the distribution of heat in fibered magnetic fields

Abstract: We study the equilibrium temperature distribution in a model for strongly magnetized plasmas in dimension two and higher. Provided the magnetic field is sufficiently structured (integrable in the sense that it is fibered by co-dimension one invariant tori, on most of which the field lines ergodically wander) and the effective thermal diffusivity transverse to the tori is small, it is proved that the temperature distribution is well approximated by a function that only varies across the invariant surfaces. The same result holds for “nearly integrable” magnetic fields up to a “critical” size. In this case, a volume of non-integrability is defined in terms of the temperature defect distribution and related the non-integrable structure of the magnetic field, confirming a physical conjecture of Paul-Hudson-Helander. Our proof crucially uses a certain quantitative ergodicity condition for the magnetic field lines on full measure set of invariant tori, which is automatic in two dimensions for magnetic fields without null points and, in higher dimensions, is guaranteed by a Diophantine condition on the rotational transform of the magnetic field.

This is joint work with Theodore D. Drivas and Dan Ginsberg.