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 (firstname.lastname@example.org) or Federico Pasqualotto (email@example.com).
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 (firstname.lastname@example.org) or Federico Pasqualotto (email@example.com).
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 (firstname.lastname@example.org) or Federico Pasqualotto (email@example.com).
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 (firstname.lastname@example.org) or Federico Pasqualotto (email@example.com).
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.