2009 Stelson Lecture - Thomas Hou
Professor Thomas Hou is the Charles Lee Powell Professor of applied and computational mathematics at the California Institute of Technology. He is also the director of the Center for Integrative Multiscale Modeling and Simulation, at Caltech. He is the recipient of many honors and awards including a Sloan fellowship, the Feng Kang Prize in Scientific Computing, and the James H. Wilkinson Prize in Numerical Analysis and Scientific Computing. Hou does research in many areas of applied mathematics including numerical analysis, large scale computations, and multi-scale phenomena. He has published over 85 scientific papers and he is the founding Editor-in-Chief of the SIAM Interdisciplinary Journal Multiscale Modeling and Simulation. Hou has presented his work at numerous conferences including as a plenary speaker at both the International Congress of Mathematicians and the International Congress on Industrial and Applied Mathematicians.
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.
Blow-up or No Blow-up? The Interplay Between Analysis and Computation in the Millennium Problem on Navier-Stokes equations
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.