This is the website of our weekly Hyperbolic & Dispersive PDE Seminar at Rutgers University. Welcome!
Organizers: Maxime Van de Moortel, Avy Soffer, Gavin Stewart.
The topics covered by the seminar are very broad and also include, among others: General Relativity, Fluid Mechanics, Harmonic Analysis, Quantum Mechanics, Spectral Theory and Applied Mathematics, always with a strong emphasis on Analysis.
The seminar in the Fall 2024 is in Hill 705, at 3:50 PM on Thursdays.
Speaker: Fabio Pusateri University of Toronto
Title: Long-time random solutions for the water waves equations
Abstract: We will present some recent results about the long-time regularity of solutions of the full gravity water waves system for initial data with random i.i.d. gaussian phases. The main new mechanism we employ is a combination of sharp deterministic energy estimates for quasilinear equations, and probabilistic and combinatorial arguments based on Feynman trees expansions. The motivation for the construction of such random solutions comes from the theory of Wave Turbulence put forth by Hasselmann in the 60s. This theory aims to describe free surface hydrodynamics, which is mathematically modelled by the water waves equations, via effective/macroscopic kinetic-type equations. In the context of the cubic NLS equation, the derivation of the wave kinetic equation has been rigorously proved in a series of recent works by Deng-Hani. Our present work initiates the rigorous analysis of Wave Turbulence in the context of the full fluids equations. This is joint work with Yu Deng (U Chicago) and Alex Ionescu (Princeton).
Speaker: Warren Li Princeton University
Title: BKL bounces outside homogeneity
Abstract: In several works spanning the late 20th century, physicists Belinski, Khalatnikov and Lifshitz (BKL) proposed a general ansatz for solutions to the Einstein equations near ‘’singularities’’, where they propose that the near-singularity dynamics at different spatial points on the singularity decouple and are approximated by solutions of a system of nonlinear autonomous ODEs. In this talk, we give a brief overview of these heuristics and present recent results justifying their validity in a large class of inhomogeneous, albeit symmetric spacetimes, which exhibit spatial decoupling even in the presence of nonlinear heteroclinic orbits of the ODE system, known as BKL bounces.