Analysis Seminar

We address the inviscid limit for the Navier-Stokes equations in a half space, with initial datum that is analytic only close to the boundary of the domain, and has finite Sobolev regularity in the complement. We prove that for such data the solution of the Navier-Stokes...

We describe a recent construction of self-similar blow-up solutions of the incompressible Euler equation. A consequence of the construction is that there exist finite-energy $C^{1,a}$ solutions to the Euler equation which develop a singularity in finite time for some range...

I will describe joint work with Boris Solomyak, in which we show that the stationary (Furstenberg) measure on the projective line associated to 2x2 random matrix products has the "correct" dimension (entropy / Lyapunov exponent) provided that the matrices satisfy a certain...

An important question in hydrodynamic turbulence concerns the scaling proprties in the inertial range. Many years of experimental and computational work suggests---some would say, convincingly shows---that anomalous scaling prevails. If so, this rules out the standard...

I will discuss random field Ising model on $Z^2$ where the external field is given by i.i.d. Gaussian variables with mean zero and positive variance. I will present a recent result that at zero temperature the effect of boundary conditions on the magnetization in a finite...

In this talk, I will give an overview of some of what is known about solutions to the thin obstacle problem, and then move on to a discussion of a higher regularity result on the singular part of the free boundary. This is joint work with Xavier Fernández-Real.

We consider a system of two interacting one-dimensional quasiperiodic particles as an operator on $\ell^2(\mathbb Z^2)$. The fact that particle frequencies are identical, implies a new effect compared to generic 2D potentials: the presence of large coupling localization...

I will discuss some new results for gradient field models with uniformly convex potentials. A connection between the scaling limit of the field and elliptic homogenization was introduced more than twenty years ago by Naddaf and Spencer. In joint work with Wei Wu, we take...

A celebrated theorem of C.L. Siegel from 1929 shows that the multiplicity of eigenvalues for the Laplace eigenfunctions on the unit disk is at most two. More precisely, Siegel shows that positive zeros of Bessel functions are transcendental. We study the fourth...

We discuss two old questions of Landis concerning behavior of solutions of second order elliptic equations. The first one is on propagation of smallness for solutions from sets of positive measure, we answer this question and as a corollary prove an estimate for...