Analysis & PDE
Organizers: Jonathan Luk & Warren Li
Past Events
Abstract: I will survey some recent developments in the theory of maximal functions that arise from averaging over families of curves in the plane. A central question in this field is to understand the L^p mapping properties of these maximal functions, and how this relates to the geometry of the…
We develop an equivariant index theorem for a twisted Dirac operator on a compact globally hyperbolic spacetime with spacelike boundary on which a group acts isometrically, subject to the Atiyah-Patodi-Singer boundary condition. Our analysis shows that the geometric formula is the same as in the…
Abstract: We present a series of works, joint with J. Bedrossian, S. He, F. Wang, in which we prove nonlinear inviscid damping, enhanced dissipation, and inviscid limit for the 2D Navier-Stokes equations near Couette. The domain is the periodic channel, \mathbb{T} \times [-1,1], and…
Abstract: Diffusion processes beyond Brownian motion have recently attracted significant interest from different communities in mathematics, the physical and biological sciences. They are described by nonlocal operators with singular non-integrable kernels, such as fractional Laplacians. The…
The Euler-Poisson system of partial differential equations describes the dynamics of a self-gravitating gas. For the energy-critical polytropic pressure law, there is an explicit steady-state solution describing an isolated star. I will discuss recent work which describes the nonlinear phase…
Abstract: We consider the focusing wave equation for all powers in all dimensions. It is well-known that the equation admits spatially homogeneous blow-up solutions, often dubbed ODE blow-up, terminating in a singular hypersurface at {t=T}. In this talk, we show both that we can construct…
Abstract: We prove time decay for the linearized beta-plane equationnear shear flow on the plane. Specifically, we show that the profilesof the velocity field components decay polynomially on any compactset, and identify specific rates of decay. Our proof entails theanalysis of oscillatory…
Suppose we are given a globally hyperbolic spacetime (M,g) solving the Einstein vacuum equations, and a timelike geodesic in M. I will explain how to construct, on any compact subset of M, a solution g_\epsilon of the Einstein vacuum equations which is approximately equal to g far from the…
The scalar model of flat bands is a simplification of models in condensed matter physics. It allows the study of relevant spectral problems using a 2nd order scalar equation, akin to the Schrödinger equation with the square of dbar on a torus replacing the Laplacian. It displays many features of…
Starting with the work of Choptuik '92, numerical relativity predicts that naked singularity spacetimes arise on the threshold of dispersion and black hole formation, a phenomenon referred to as critical collapse. In this talk, I will present for 2+1 gravity the first rigorous construction of…