Analysis Seminar - SPRING 2005

Analysis / PDE Seminar (Spring 2005)

Time: 2:40-3:30pm Wednesdays Room: PSA 103 (unless otherwise specified) Contact: Svetlana Roudenko - svetlana [at] math.asu.edu Fall 2004
DATE	SPEAKER	TOPIC/ABSTRACT
January 26
February 2	Prof. Anne Gelb, ASU	Title: Spectral Viscosity Approximations for Shallow Water Equations on Spheres Abstract: We discuss spectral methods for hyperbolic partial differential equations, notably Fourier, Legendre and spherical harmonic approximations. In solving nonlinear equations, shocks can form in the solutions, causing the well known Gibbs oscillations to appear near the discontinuities and an overall reduced order of accuracy. Such oscillations, if left unchecked, will contaminate the solution in the entire domain and cause instabilities. On the other hand, strong viscosity will eliminate the small scales or finer features of the solution. Spectral viscosity methods have been created to add viscosity only at the tip of the spectrum, allowing the underlying solution to retain its spectral convergence properties. With proper post-processing, spectral accuracy can still be realized. Several examples including shallow water equations on spheres are considered.
February 11, (*Friday* ) time : 2:40-3:30pm room : PSA 109	Dr. Lotfi Hermi, University of Arizona	Title: A Highlight of New and Classical Eigenvalue Inequalities Abstract: There is an analogy between inequalities involving the base frequency of a fixed membrane and those involving the (scaled) volume for a free membrane. We exploit this observation to prove two new low bounds of Weyl-type for the higher pure tones of a drum. The talk will trace some of these inequalities since the classical works of Weyl (1910) and Payne-Polya-Weinberger (1956) and describe more recent results by Ashbaugh-Benguria, Harrell, Friedlander, Laptev and others.
February 16	Dr. Juergen Saal, Vanderbilt Univ. & Darmstadt Univ. of Technology, Germany	Title: Stokes and Navier-Stokes Equations with Robin Boundary Conditions in a Half-Space for Lebesgue-Integrable or Bounded Initial Data
February 21, (Monday) time : 3:40-4:30pm room : PSF 101 (refreshments at 3pm in PSA 206)	Distinguished Series Colloquium: Prof. Carlos Kenig, Louis Block Professor of Mathematics and the College, University of Chicago	Title: Recent progress on the well-posedness of nonlinear dispersive equations Abstract: We will survey some of the developments in the subject from the 90's and discuss some important problems which fall outside of their scope. Some recent partial progress in these problems will be reported.
February 22, (Tuesday) time : 3:30-4:30pm room : PSA 111 (refreshments at 3pm in PSA 206)	Special Seminar: Prof. Carlos Kenig, Louis Block Professor of Mathematics and the College, University of Chicago	Title: Anderson localization for the continuous Bernoulli model and quantitative unique continuation Abstract: We will discuss the recent proof ( joint with Bourgain ) of Anderson localization for the continuous Anderson-Bernoulli model, emphasizing its connection with quantitative uniquec ontinuation problems.
March 2	Prof. John McDonald, ASU	Title: Phase retrieval problems (continued) Abstract: We continue with a discussion of the phase retrieval problem in one dimension, i.e., given functions f and g on the line such that \|f\|=\|g\|, what further conditions force g=cf for some constant. In this talk we consider some new results on the phase retrieval problem in the case where f and g belong to the classical Hardy space H^2. Time permiting we will also touch on the problem resolving of Pauli pairs (two L_2 functions satisfying \|f\|=\|g\| and \|F(f)\|=\|F(g)\|, where F is the Fourier transform.)
March 9	Dr. Elena Frolova, St. Petersburg Electrotechnical University, Russia	Title: L_p theory for Stefan and Verigin problems Abstract: We consider free boundary problems for second order parabolic equations. We present local solvability results for Stefan and Verigin problems in anisotropic Sobolev spaces. There is no loss of regularity. We will discuss the proof of L_p estimates for model half-space problems for parabolic and elliptic equations with time derivatives in the boundary or conjugation conditions.
March 11 *(Friday)* time : 2:40-3:30pm room : PSA 106	Dr. Sergey Repin, Steklov Mathematical Institute, St Petersburg, Russia	Title: A posteriori estimates for approximate solutions of elliptic type PDE's Abstract: An overwiew of existing methods of a posteriori error control is presented. Main attention is paid on new type estimates (functional type a posteriori estimates) developed in the last decade. They are considered for the following typical boundary--value problems: diffusion equation (linear elliptic equation), Stokes problem (elliptic problem on a subspace of divergent--free functions), obstacle problem (elliptic type variational inequality), Bingham fluid model (nonlinear variational problem on a subspace).
March 16	Spring Break
March 23	Dr. Timofey Shilkin, Steklov Mathematical Institute, St Petersburg, Russia	Title: On the coupled systems in the mathematical fluid dynamics Abstract: We discuss some systems of PDEs arising in the mathematical thermo- and magnetohydrodynamics. Typically, these system involve the Navier Stokes-type equations coupled by some other nonlinear parabolic equations. In particular, for the partialy viscous 2D- MHD system the regorous mathematical results on the existence and uniqueness of a global strong solution will be presented.
March 24 *(Thursday)* time : 3:40-4:30pm room : PSA 102	Special Colloquium: Dr. Gregory Seregin, Steklov Mathematical Institute, St Petersburg, Russia	Title: Weak solutions to the Cauchy problem for the 3D Navier-Stokes equations satisfying the local energy inequality Abstract: We discuss the existence of weak solutions to the Cauchy problem for the 3D Navier-Stokes equations with initial data belonging to the special Morrey class $L_{2,unif}$. These solutions satisfy the local energy inequality. For this reason, the Caffarelli-Kohn-Nirenberg theory of partial regularity is applicable to them.
March 30	Dr. Timofey Shilkin, Steklov Mathematical Institute, St Petersburg, Russia	Title: On the coupled systems in the mathematical fluid dynamics (cont.) Abstract: We continue with the partially viscous 2D - MHD system and prove the existence and uniqueness of a global strong solution.
April 4 (Monday) time : 1-2pm room : PSF 101	Special Seminar: Dr. Rodrigos Banuelos, Purdue University (joint with Cesar Chavez Celebration Events)	Title: Martingales and Singular Integrals Abstract: The Beurling-Ahlfors operator in the complex plane and its generalizations to differentials forms in several dimensions are examples of Calderón-Zygmund singular integrals. They play a fundamental role in the study of quasiconformal maps as evident in the seminal papers of Donalson and Sullivan (1989) ``Quasiconformal 4 -manifolds" and Iwaniec and Martin (1993) ``Quasiconformal mappings in even dimensions." An open problem of considerable importance (known as Iwaniec's Conjecture) is the identification of the L^p-norms of these operators. These norms provide information on the regularity of quasiconformal maps and have implications for the existence of minimizers of conformally invariant functionals in the calculus of variations. Unfortunately, the Calderón-Zygmund theory, as powerful as it is, does not provide the desired information on these norms. In this talk we will describe how techniques from probability theory, particularly from martingale theory, can be used to obtain very precise estimates for these operators. This talk is designed for a general audience. Technicalities will be kept to a minimum while at the same time providing a flavor of the ideas involved in this approach.
April 13, time : 6pm room : PSF 173	Special Colloquium: (joint with Math Awareness Month) Prof. Arlie Petters, Duke University	Title: GRAVITY'S COSMIC MIRAGES: A Mathematical Unveiling Abstract: Galaxies can act like lenses, creating cosmic mirages. One such mirage is the occurrence of multiple images of distant light sources, an effect predicted by Einstein in 1936. Strikingly, these mirages allow us to probe one of the mysteries of the universe - the nature of dark matter. Though dark matter is invisible, it can be revealed through its action as a gravitational lens. This talk gives an introduction to the vibrant subject of gravitational lensing and a tour of the mathematical underpinnings of these cosmic mirages.
April 20	Dr. Okihiro Sawada, Waseda University, Tokyo, Japan	Title: On the time-local existense of the Navier-Stokes flows with linearly growing initial data
April 27	Special Seminar: Prof. Alexander Volberg, Distinguished University Professor, Michigan State University (Salem Prize 1999)	Title: Some estimates of combination of Riesz transforms and Bellman function technique.