The APDE seminar on Monday, 5/3, will be given by Elena Giorgi online via Zoom from 4.10pm to 5.00pm PT. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: The stability of charged black holes.

Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.

The APDE seminar on Monday, 4/26, will be given by Rita Teixeira da Costa online via Zoom from 12.10pm to 1.00pm PT (note the time change). To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Oscillations in wave map systems and an application to General Relativity

Abstract: Due to their nonlinear nature, the Einstein equations are not closed under weak convergence. Compactness singulaties associated to highly oscillatory solutions may be identified with some non-trivial matter. In 1989, Burnett conjectured that, for vacuum sequences, this matter produced in the limit is captured by the Einstein-massless Vlasov model. 
In this talk, we give a proof of Burnett’s conjecture under some gauge and symmetry assumptions, improving previous work by Huneau—Luk from 2019. Our methods are more general, and apply to oscillating sequences of solutions to the wave maps equation in (1+2)-dimensions.
This is joint work with André Guerra (University of Oxford).

The APDE seminar on Monday, 4/19, will be given by Namaluba Malawo online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Resonances for thin barriers on the half-line.

Abstract: The analysis of scattering by thin barriers is important for many physical problems, including quantum corrals. To model such a barrier, we use a delta function potential on the half-line. Our main results compute decay rates for particles confined by this barrier. The decay rates are given by imaginary parts of resonances. We show that they energy dependence of the decay rates is logarithmic when the barrier is weaker and polynomial when the barrier is stronger. To compute them, we derive a formula for resonances in terms of the Lambert W function and apply a series expansion. Joint work with Kiril Datchev.

The APDE seminar on Monday, 4/12, will be given by Jeffrey Kuan online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: A stochastic fluid-structure interaction model given by a stochastic viscous wave equation

Abstract: We consider a stochastic fluid-structure interaction (FSI) model, given by a stochastic viscous wave equation perturbed by spacetime white noise. The wave equation part of the model describes the elastodynamics of a thin structure, such as an elastic membrane, while the viscous part, which is in the form of the Dirichlet-to-Neumann operator, describes the impact of a viscous, incompressible fluid in a two-way coupled fluid-structure interaction problem. The stochastic perturbation describes random deviations observed in real-life data. We prove that this stochastic viscous wave equation has a mild solution in dimension one, and also in dimension two, which is the physical dimension of the FSI problem (thin 2D membrane). This behavior contrasts that of the stochastic heat and the stochastic wave equations, which do not have function valued mild solutions in dimensions two and higher. This means that in the two dimensional model, unlike the heat and wave equations, dissipation due to fluid viscosity in the viscous wave equation, keeps the stochastically perturbed solution “in control”. We also consider Hölder continuity path properties of solutions and show that the solution is Hölder continuous up to Hölder exponent 1/2 in both space and time, after stochastic modification. This is joint work with Suncica Canic.

The APDE seminar on Monday, 4/5, will be given by Maciej Zworski online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Internal waves and homeomorphism of the circle.

Abstract: The connection between the formation of internal waves in fluids and
the dynamics of homeomorphisms of the circle was investigated by
oceanographers in the 90s and resulted in novel experimental
observations (Maas et al, 1997). The specific homeomorphism is given
by a chess billiard” and has been considered by many authors (John
1941, Arnold 1957, Ralston 1973, … , Lenci et al 2021). The relation
between the nonlinear dynamics of this homeomorphism and linearized
internal waves provides a striking example of classical/quantum
correspondence (in a classical and surprising setting of fluids!) and,
using a model of tori and of zeroth order pseudodifferential
operators, it has been a subject of recent research, first by Colin de
Verdière-Saint Raymond 2020 and then by Dyatlov, Galkowski, Wang and
the speaker. In these works, many facets of the relationship between
hyperbolic sources and sinks for the classical dynamics and internal
waves in fluids were explained. I will present some of these results
as well as some numerical discoveries (including those of
Almonacid-Nigam 2020). I will also describe various open problems.

The APDE seminar on Monday, 3/29, will be given by Larry Guth online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Local smoothing for the wave equation.

Abstract: The local smoothing problem asks about how much solutions to the wave equation can focus. It was formulated by Chris Sogge in the early 90s. Hong Wang, Ruixiang Zhang, and I recently proved the conjecture in two dimensions.
In this talk, we will build up some intuition about waves to motivate the conjecture, and then discuss some of the obstacles and some ideas from the proof.

The APDE seminar on Monday, 3/8, will be given by John Anderson online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Stability results for anisotropic systems of wave equations

Abstract: In this talk, I will describe a global stability result for a nonlinear anisotropic system of wave equations. This is motivated by studying phenomena involving characteristics with multiple sheets. For the proof, I will describe a strategy for controlling the solution based on bilinear energy estimates. Through a duality argument, this will allow us to prove decay in physical space using decay estimates for the homogeneous wave equation as a black box. The final proof will also require us to exploit a certain null condition that is present when the anisotropic system of wave equations satisfies a structural property involving the light cones of the equations.

The APDE seminar on Monday, 2/22, will be given by Alexis Drouot online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Mathematical aspects of topological insulators.

Abstract: Topological insulators are intriguing materials that block conduction in their interior (the bulk) but support robust asymmetric currents along their edges. I will discuss their analytic, geometric and topological aspects using an adiabatic framework favorable to quantitative predictions.

The APDE seminar on Monday, 2/8, will be given by Kihyun Kim online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Blow-up dynamics for the self-dual Chern-Simons-Schrödinger equation

Abstract: We consider the blow-up dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariance symmetry. (CSS) is $L^2$-critical, has the pseudoconformal symmetry, and admits a soliton $Q$ for each equivariance index $m \geq 0$. An application of the pseudoconformal transformation to $Q$ yields an explicit finite-time blow-up solution $S(t)$ which contracts at the pseudoconformal rate $|t|$. In the high equivariance case $m \geq 1$, the pseudoconformal blow-up for smooth finite energy solutions in fact occurs in a codimension one sense, but also exhibits an instability mechanism. In the radial case $m=0$, however, $S(t)$ is no longer a finite energy blow-up solution. Interestingly enough, there are smooth finite energy blow-up solutions whose blow-up rates differ from the pseudoconformal rate by a power of logarithm. We will explore these interesting blow-up dynamics (with more focus on the latter) via modulation analysis. This talk is based on my joint works with Soonsik Kwon and Sung-Jin Oh.

The APDE seminar on Monday, 11/30, will be given by Khang Manh Huynh online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Construction of the Hodge-Neumann heat kernel, local Bernstein estimates, and Onsager’s conjecture in fluid dynamics.

Abstract: Most recently, in arXiv:1907.05360 [math.AP], we introduced the theory of heatable currents and proved Onsager’s conjecture on Riemannian manifolds with boundary, where the weak solution has $B_{3,1}^{\frac{1}{3}}$ spatial regularity. In this sequel, by applying techniques from geometric microlocal analysis to construct the Hodge-Neumann heat kernel, we obtain off-diagonal decay and local Bernstein estimates, and then use them to extend the result to the Besov space $\widehat{B}_{3,V}^{\frac{1}{3}}$, which generalizes both the space $\widehat{B}_{3,c(\mathbb{N})}^{1/3}$ from arXiv:1310.7947 [math.AP] and the space $\underline{B}_{3,\text{VMO}}^{1/3}$ from arXiv:1902.07120 [math.AP] — the best known function space where Onsager’s conjecture holds on flat backgrounds.