Department of Mathematics, University of Kentucky Analysis and PDE Seminar

Fall 2010

Partially supported by the Enrichment Fund of the College of Arts and Sciences and by the Department of Mathematics.

Date SPEAKER and TITLE Host
September 7
NO SEMINAR SCHEDULED

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September 14 Professor Michael Frazier, University of Tennessee
TITLE: Global Estimates for Kernels of Neumann Series, Green's Functions, and the Conditional Gauge
ABSTRACT: pdf file
September 21 Professor Peter Perry, University of Kentucky
TITLE: Global Well-Posedness for the Davey-Stewartson Equation in $H^{1,1}(R^2)$
ABSTRACT: pdf file
Katy Ott
September 28 Professor John Lewis, University of Kentucky
TITLE: On the Dimension of p-Harmonic Measure in Space
ABSTRACT: pdf file
Katy Ott
October 5 Professor David Adams, University of Kentucky
TITLE: Morrey Spaces in Harmonic Analysis (HA)
ABSTRACT: pdf file
Katy Ott
October 12
NO SEMINAR SCHEDULED

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October 19
NO SEMINAR SCHEDULED

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October 26
NO SEMINAR SCHEDULED

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November 2 Professor Dehua Wang, University of Pittsburgh
TITLE: Transonic flow and isometric embedding
ABSTRACT: Some recent research progress on the multi-dimensional compressible Euler equations will be reviewed. In particular, the transonic flows past an obstacle such as an airfoil, and the isometric embeddings in geometry via a fluid dynamics approach will be discussed. The talk is based on the joint works with Gui-Qiang Chen and Marshall Slemrod.
Changyou Wang
November 11 **Special Seminar (Thursday), Qualifying Examination**
Mr. Jay Hineman, University of Kentucky
TITLE: On Leray's solutions to the Navier-Stokes equation in $R^3$
ABSTRACT: I will summarize the main results of Leray's landmark 1934 Acta mathematica paper: "Sur le mouvement d'un liquide visqueux emplissant l'space" [On the motion of a viscous liquid filling space]. Firstly, I will discuss Leray's method for proving the short-time existence of strong solutions, the related criteria for finite-time singularity and the uniqueness of such solutions. Secondly, I will discuss the existence of Leray's weak solutions and a weak-strong uniqueness principle. The uniqueness and regularity of Leray's weak solutions remain open problems.
Changyou Wang
November 16
** Qualifying Examination **
Mr. Tao Huang, University of Kentucky
TITLE: $L_{3,\infty}$-solutions of the Navier-Stokes Equations and Backward Uniqueness
ABSTRACT: I will discuss the proof of $L_{3,\infty}$-solutions of three dimensional Navier-Stokes equations are smooth. I will also discuss the proof of backward uniqueness of parabolic equation on $\mathbb{R}^n\setminus B(R)\times(0,T)$ by using Carleman type inequalities.
Changyou Wang
November 23
NO SEMINAR SCHEDULED -- Thanksgiving Break

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November 30
No seminar scheduled

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December 7 Professor Abel Klein, UC Irvine
TITLE: Universal occurrence of localization in continuous random Schr\"odinger operators
ABSTRACT: We prove a strong form of localization at the bottom of the spectrum for continuous Anderson Hamiltonians with non-degenerate single site probability distribution of bounded support, without any additional condition on the single site probability distribution. This strong form of localization includes Anderson localization (pure point spectrum with exponentially decaying eigenfunctions), dynamical localization (no spreading of wave packets under the time evolution), and log-Holder continuity of the integrated density of states.
Peter Hislop
December 14 *** Location changed to POT 845 ***
Professor Zhongwei Shen , University of Kentucky
TITLE: Homogenization of Elliptic Systems with Neumann Boundary Conditions
ABSTRACT: In this talk I will discuss my recent joint work with Carlos Kenig and Fanghua Lin on uniform regularity estimates for a family of second order elliptic systems, arising in the theory of homogenization, with rapidly oscillating periodic coefficients. We obtain sharp W^{1,p} estimates, Lipschitz estimates, and nontangential maximal function estimates on solutions with Neumann boundary conditions in C^{1,\alpha} domains. The problem has been open for more than 20 years, ever since the analogous results for solutions with the Dirichlet conditions were established by Avellaneda and Lin in late 80’s.
Katy Ott