The Analysis Seminar

Spring 2012

The seminar meets Wednesdays in room LN 2205 at 3:30 p.m. There are refreshments and snacks in the Anderson Reading Room at 3:15.

Organizer: Paul Loya

To receive announcements of seminar talks by email, please join the Analysis Seminar's mailing list.

February 15: Ye Li (Binghamton)
Title: Introduction to Schoen and Yau's book, Lectures on Differential Geometry
Abstract: I will present a series of talks focusing on Yau's gradient estimates for elliptic equations on Riemannian manifolds with lower bounds of Ricci curvature. This now becomes one of the fundamental techniques in geometric analysis. It has many applications, such as Harnack inequalities, eigenvalue estimates, Cheeger-Colding theory, etc. We will follow Schoen and Yau's book, Lectures on Differential Geometry. The first talk will be elementary. We will discuss manifolds, Riemannian metrics, connections, curvature tensors, etc.

February 21 (special time and day; 10:05-11:05): Chi Hin Chan (The Institute of Mathematical Sciences, The Chinese University of Hong Kong)
Title: About the regularity of solutions to an evolution equation with a type of nonlocal integral operator.
Abstract: In this talk, we will present a piece of work with Luis Caffarelli and Alexis Vasseur, in which we establish the Holders' regularity of solutions to a parabolic equation with a type of nonlocal integral operator. The type of nonlocal integral operator on which we perform our analysis is the one with the kernel K(t,x,y) comparable to that of a fractional Laplacian in the following sense: K(t,x,y) is globally bounded above (for any x, y) and locally bounded below (when x,y satisfies |x-y| < 4 ) by some suitable scalar multiples of the kernel of a fractional Laplacian. According to the above description of the kernel K(t,x,y), the associated nonlocal integral operator that we chose will preserve the local regularization effect of a standard fractional Laplacian in a parabolic equation. However, since the translation-invariance structure of K(t,x,y) is lacking (not available), our argument, which eventually leads to the Holders' regularity of solutions to our parabolic equation with the above mentioned nonlocal integral operator, is surprisingly delicate and is based on the De Giorgi's method.

February 29: Daniel da Silva (Rochester)
Title: Global regularity in generalized wave maps
Abstract: Wave maps are nonlinear generalizations of the wave equation which have been studied for decades. In this talk, we will consider generalizations of wave maps based on the Skyrme and Adkins-Nappi models of nuclear physics. These models yield nonlinear hyperbolic partial differential equations, for which we consider the question of regularity of solutions. In particular, we will discuss the non-concentration of energy in these models, a preliminary step in establishing a global regularity theory.

March 7: Marcelo Mendes Disconzi (Stony Brook)
Title: Motion of Slightly Compressible Fluids in a Bounded Domain.
Abstract: We study the initial-boundary value problem for equations of inviscid fluid motion in a bounded domain in R^n. We show that the solution to this problem for a slightly compressible fluid (or fluid with high sound speed) is near to that of an incompressible fluid. We also prove that the solution to the initial-value problem depends in a $C^1$ fashion on the initial data. Such a dependence is unusual for non-linear equations. This is a joint work with David Ebin.

March 21: Dmitri Scheglov (Oklahoma)
Title: Triangle billiard complexities
Abstract: We discuss several results related to the growth of orbits for a triangle billiard.

March 28: Ye Li (Binghamton)
Title: Introduction to Schoen and Yau's book, Lectures on Differential Geometry. Part 2.
Abstract: In the second talk of discussing Schoen and Yau's Lectures on Differential Geometry, I will focus on the Bochner identity and its applications in geometry, in particular, Cheeger and Gromoll's splitting theorem on the Riemannian manifold with non-negative Ricci curvature.

April 18: Anna Mazzucato (Penn State)
Title: Boundary layer analysis for certain classes of non-linear incompressible flows
Abstract: We present recent results on the analysis of the vanishing viscosity limit and associated boundary layer in certain classes of non-linear 3D flows in pipes and channels. We use both effective equations for flow correctors and singular perturbation analysis for a heat equation with variable drift.

April 25: Marian Gidea (Northeastern Illinois University and IAS)
Title: Instability and Diffusion in Hamiltonian Systems
Abstract: The Arnold diffusion problem, dating from 1964, conjectures that generic mechanical systems exhibit `diffusing' trajectories, which explore a large fraction of the phase space over time. Equivalently, the total energy of a generic mechanical system can be made arbitrarily large by applying small, time-dependent perturbations to the system. We consider two models for the Arnold diffusion problem: the perturbed geodesic flow and the large gap problem. We discuss some geometric and topological mechanisms that determine the existence of diffusing orbits in these systems. The geometric mechanisms are based on normally hyperbolic invariant manifolds, KAM tori, Aubry-Mather sets, and on the heteroclinic connection among these objects. The topological mechanisms are based on Conley index theory -- in particular, on the topological method of correctly aligned windows -- and on the obstruction property of invariant manifolds. The interplay of these mechanisms yields explicit methods for detecting diffusing orbits. Such methods can be used to obtain quantitative estimates on the diffusing orbits, and can be implemented in computer assisted proofs. Moreover, they apply to larger classes of examples when compared to geometric perturbation theory or variational methods.

May 2: Ye Li (Binghamton)
Title: Introduction to Schoen and Yau's book, Lectures on Differential Geometry, Part 3
Abstract: In the third talk of discussing Schoen - Yau's book, Lectures on Differential Geometry, I will focus on the Laplacian and volume comparison theorems, which plays an important role in Yau's gradient estimate. Also these two comparison theorems are essential in Cheeger-Colding's works on almost rigidity theorems in Riemannian manifolds with Ricci curvature bounded from below.

May 9: Ye Li (Binghamton)
Title: Introduction to Schoen and Yau's book, Lectures on Differential Geometry, Part 4.
Abstract: In the final talk of discussing Schoen - Yau's book, Lectures on Differential Geometry, I will focus on Yau's gradient estimate, which now is a standard technique in geometric analysis. To have a better understanding of the result, I will first consider the Euclidean case, i.e., Bernstein's estimate.