Computational and Applied Math Proseminar

Friday, November 14, 2008, 2:00 p.m. PSA 118

Robert Finn

Dept Math, Stanford University

Magnitude and gradient estimates for stationary solutions of the Navier-Stokes equations; a computer-aided study

Abstract We calculate numerically solutions of the stationary Navier-Stokes equations in two dimensions, for a square domain with particular choices of boundary data. The data are chosen to test whether bounded disturbances on the boundary can be expected to spread into the interior of the domain. The results indicate that such behavior indeed can occur, but suggest an estimate of general form for the magnitudes of the solution, analogous to classical bounds for harmonic functions. A corresponding estimate for the derivatives would then follow from earlier work by Finn and Solonnikov. The qualitative behavior of the solutions we found displayed some striking and unexpected features. As a corollary of the study, we obtain two new examples of non-uniqueness for stationary solutions at large Reynolds number.

This is joint work with A. Quazzi and S. Turek.

Short Bio of Robert Finn: As a graduate student at Syracuse University, Finn had the honor of taking a course from Paul Erdos. (When he couldn't solve one of the homework problems but could only reduce it to a lemma he couldn't prove, Erdos responded "In Hungary, when a six year old boy doesn't know that lemma, we throw him in the Danube." Since that experience, Finn's swimming has improved.) Professor Finn earned a PhD in Mathematics from Syracuse in 1951 and held visiting positions at the Institute for Advanced Study and the University of Maryland and then served as an assistant professor at the University of Southern California, associate professor at the Calfornia Institute of Technology and, starting in 1959, professor at Stanford University. He has published approximately 160 papers in the areas of fluid mechanics, partial differential equations and the calculus of variations. He serves on the editorial board of several journals and is vice-president of the Pacific Journal of Mathematics.

Robert Finn is the leading expert in the mathematical theory of capillarity. Paul Concus and Bob Finn proposed that NASA conduct capillary experiments in space to test the validity of the unusual predictions of (mathematical) capillary theory, which is important to the design of fuel tanks for vehicles in space and plumbing in space stations and vehicles. Together with Mark Weislogel, Concus and Finn designed experiments which were conducted in NASA drop towers, the space shuttle (two experiments) and the Russian MIR space station.

For further information please contact: mittelmann@asu.edu