Harnack inequalities for degenerate and singular parabolic operators

Speaker: 

Prof. Vincenzo Vespri

Institution: 

Universita' degli Studi di Firenze

Time: 

Friday, May 2, 2008 - 4:00pm

Location: 

MSTB 254

Parabolic Harnack inequalities were proved by Moser for linear equation with bounded and measurable coefficients. In the case of the parabolic p-Laplacean such kind of estimates cannot hold (as proved by Trudinger). In the nineties DiBenedetto introduced the so called intrinsic Harnack inequalities for the protype equation. His original proof requiries the maximum principle and the existence of suitable subsolutions. Therefore the proof for general equations (with bounded and measurable coefficient) was missing. In some recent papers, in collaboration with DiBendetto and Gianazza, we proved intrinsic Harnack inequalities for general degenerate and singular operators. We show, via suitable counterexamples, that such estimates are sharp. Moreover we proved that when p is approaching to 2, our estimates tend to the classical Moser estimates.

Fast Multiscale Clustering and Manifold Identification.

Speaker: 

Dan Kushnir

Institution: 

Weizmann Institute of Science

Time: 

Tuesday, February 19, 2008 - 3:00pm

Location: 

MSTB 254

I will present a novel multiscale clustering algorithm inspired by algebraic multigrid techniques. Our method begins with assembling
data points according to local similarities. It uses an aggregation process to obtain reliable scale-dependent global properties, which arise from the local similarities. As the aggregation process proceeds, these global properties influence the formation of coherent clusters. The
global features that can be utilized are for example density, shape, intrinsic dimensionality and orientation. The last three features are a
part of the manifold identification process which is performed in parallel to the clustering process. The algorithm detects clusters that
are distinguished by their multiscale nature, separates between clusters with different densities, and identifies and resolves intersections between clusters. The algorithm is tested on synthetic and real data sets, its running time complexity is linear in the size of the data set.

Joint work with: Meirav Galun and Achi Brandt.

The Primitive Equations in Two Space Dimensions With Multiplicative Noise

Speaker: 

Nathan Glatt-Holtz

Institution: 

University of Southern California

Time: 

Tuesday, February 5, 2008 - 3:00pm

Location: 

MSTB 254

The Primitive Equations are a fundamental model describing large scale oceanic and atmospheric processes. They are derived from the fully compressible Navier-Stokes equations on a combined basis of scale analysis and meteorological data. While an extensive body of mathematical literature exists in the study of these systems, very little is known in the stochastic setting. In this talk we discuss recent joint work with M. Ziane concerning existence and uniqueness of solutions for the 2-D equations in the presence of multiplicative noise terms.

Some recent results on the two-layer quasi-geostrophic beta plane equations.

Speaker: 

Professor Lee Panetta

Institution: 

Texas A & M University

Time: 

Friday, January 11, 2008 - 4:00pm

Location: 

MSTB 254

The two-layer beta-plane quasi-geostrophic (QG) model plays a central role in theoretical studies of atmospheric and oceanic dynamics. It is a pair of coupled non-linear partial differential equations involving functions of two space variables and one time variable (streamfunctions for coupled two-dimensional flows). Solutions represent flows in a sense intermediate between 2-d and 3-d flows: they have a mild form of the ``vortex stretching'' process, absent in 2-d flows, that is at the heart of the difficulty in proving the long-time existence of classical solutions to the
3-d Navier-Stokes equations.

Numerical solutions to these QG equations display analogues of important features of atmospheric and oceanic flow, some of which I will illustrate. As is true of climate models, many interesting features are revealed only by long time averaging of the numerical solutions. The results I will present, on long-time existence of regular solutions and on dissipativity,
are part of an effort to provide a rigorous justification for this averaging, something beyond our reach in the case of the vastly more complicated climate models.

The talk will place the model in the context of other QG models, point out a useful formal similarity to the Kuramoto-Sivishinsky equation, and sketch proofs of the main results. The work is joint with C. Foias, C. Onica, E. Titi, and M. Ziane.

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