The Analysis Seminar

Fall 2011

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

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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.