Brown Analysis Seminar
Abstract: Hermite Riesz transforms are singular operators, whose $L^p$ estimate is intimately related to Hilbert transform along parabolas. Therefore one would expect that the estimate of their norm as operators in $L^p$ cannot be linear (this is "expected", not proved, but the rationale is strong based on papers of Seeger, Tao and Wright on the behavior of such operators near $L^1$). Unexpectedly, Bellman function approach gives a linear in $p$ estimate of Hermite Riesz transforms. This is a joint work with Oliver Dragicevic.
Boston University/Brown University PDE Seminar
Abstract: In this talk, I will consider long time asymptotics for hyperbolic-parabolic system of conservation laws. These equations govern the time evolutions of many physical systems in which both transport properties and diffusion are present, such as gas dynamics, magneto-hydro-dynamics and the like. It was previously known that first order asymptotics were mainly of parabolic type, and I will explain how the nonlinear interaction of those leading order diffusion waves necessary produce slowly decaying (in both space and time) higher order corrections. In sharp contrast with single conservation laws, this phenomenon occurs even for initial conditions of compact support. I will also explain how to reconcile these (seemingly) long tails with the conservation of finiteness for the spatial moments, and how these observations beg for the development of a theory mixing spatial dynamics and singular perturbation for PDE's in which spatial scales are time-dependent.
Department of Mathematics Colloquium
Abstract: These two important notions of geometric measure theory will be explained in the talk. We will show their relations with such diverse subject in Harmonic analysis as analytic capacity and estimates of operators with rough kernels. Some undercooked noodles (this is a mathematical notion not an epithet) might be prepared during the talk.
Boston University/Brown University PDE Seminar
Abstract: We give a convergent expansion of solutions of the two-dimensional, incompressible Navier-Stokes equations which generalizes the Helmholtz-Kirchhoff point vortex model to systematically include the effects of both viscosity and finite core size. The evolution of each vortex is represented by a system of coupled ordinary differential equations for the location of its center, and for the coefficients in the expansion of the vortex with respect to a basis of Hermite functions. The differential equations for the evolution of the moments contain only quadratic nonlinearities and we give explicit combinatorial formulas for the coefficients of these terms. We also show that in the limit of vanishing viscosity and core size we recover the classical Helmholtz-Kirchhoff point vortex model. This is joint work with R. Nagem, G. Sandri, and D. Uminsky.
Scientific Computing Seminar
PDE Seminar
Abstract: In this talk, we consider the Klein-Gordon-Schrodinger system with Yukawa coupling in three space dimensions. We study the orbital stability of a standing wave, which is written by the unique positive solution to a well-known scalar field equation. By applying the general theory of Grillakis et al, we show that the standing wave is stable.
<--- 2008 Index