Seminars and Colloquia by Series

Role of Mathematics Across Science and Beyond

Series
Stelson Lecture Series
Time
Monday, November 22, 2010 - 16:30 for 1 hour (actually 50 minutes)
Location
Klaus 1116
Speaker
James Glimm – University of Stony Brook, Department of Applied Mathematics and Statistics

Please Note: This lecture is more for the general audience. Reception to follow in Klaus Atrium.

The changing status of knowledge from descriptive to analytic, from empirical to theoretical and from intuitive to mathematical has to be one of the most striking adventures of the human spirit. The changes often occur in small steps and can be lost from view. In this lecture we will review vignettes drawn from the speaker's personal knowledge that illustrate this transformation in thinking. Examples include not only the traditional areas of physics and engineering, but also newer topics, as in biology and medicine, in the social sciences, in commerce, and in the arts. We also review some of the forces driving these changes, which ultimately have a profound effect on the organization of human life.

Blow-up or no Blow-up? The Interplay Between Analysis and Computation in the Millennium Problem on Navier-Stokes Equations

Series
Stelson Lecture Series
Time
Tuesday, October 27, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Thomas Y. Hou – California Institute of Technology, Applied and Computational Mathematics

Please Note: This lecture will be more for the mathematical audience

Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the seven Millennium Problems posted by the Clay Mathematical Institute. We review some recent theoretical and computational studies of the 3D Euler equations which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. The local geometric regularity of vortex filaments can lead to tremendous cancellation of nonlinear vortex stretching. This is also confirmed by our large scale computations for some of the most well-known blow-up candidates. We also investigate the stabilizing effect of convection in 3D incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. Here we reveal a surprising nonlinear stabilizing effect that the convection term plays in regularizing the solution. Finally, we present a new class of solutions for the 3D Euler and Navier-Stokes equations, which exhibit very interesting dynamic growth property. By exploiting the special structure of the solution and the cancellation between the convection term and the vortex stretching term, we prove nonlinear stability and the global regularity of this class of solutions.

Multiscale Modeling and Computation - The Interplay Between Mathematics and Engineering Applications

Series
Stelson Lecture Series
Time
Monday, October 26, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
SST Room 2
Speaker
Thomas Y. Hou – California Institute of Technology, Applied and Computational Mathematics

Please Note: This lecture is more for the general audience.  Reception following lecture. Organizers: Chongchun Zeng and Hao Min Zhou

Many problems of fundamental and practical importance contain multiple scale solutions. Composite and nano materials, flow and transport in heterogeneous porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the wide range of length scales in the underlying physical problems. Direct numerical simulations using a fine grid are very expensive. Developing effective multiscale methods that can capture accurately the large scale solution on a coarse grid has become essential in many engineering applications. In this talk, I will use two examples to illustrate how multiscale mathematics analysis can impact engineering applications. The first example is to develop multiscale computational methods to upscale multi-phase flows in strongly heterogeneous porous media. Multi-phase flows arise in many applications, ranging from petroleum engineering, contaminant transport, and fluid dynamics applications. Multiscale computational methods guided by multiscale analysis have already been adopted by the industry in their flow simulators. In the second example, we will show how to develop a systematic multiscale analysis for incompressible flows in three space dimensions. Deriving a reliable turbulent model has a significant impact in many engineering applications, including the aircraft design. This is known to be an extremely challenging problem. So far, most of the existing turbulent models are based on heuristic closure assumption and involve unknown parameters which need to be fitted by experimental data. We will show that how multiscale analysis can be used to develop a systematic multiscale method that does not involve any closure assumption and there are no adjustable parameters.

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