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ACMD Seminar: Conservation Laws, Lagrangian Dynamics, Diffeomorphisms and Weak Solutions

Barbara Lee Keyfitz
Department of Mathematics, The Ohio State University

Tuesday, May 10, 3:00-4:00 EDT (1:00-2:00 MDT)

A video of this talk is available to NIST staff in the Math channel on NISTube, which is accessible from the NIST internal home page.

Abstract: Well-known fact: The system of equations governing compressible ideal gas dynamics can be expressed in either “Eulerian” coordinates, which describe the attributes of a flow (density, velocity, energy and so on) from the point of view of a stationary observer, or in “Lagrangian” coordinates, which follow the path of a hypothetical particle traveling with the fluid. Furthermore, the two different formulations correspond perfectly, even for weak solutions (when, for example, shock waves are present) or when vacuum is present in part of the flow field.

When the flow is smooth, Lagrangian coordinates describe the evolution in time of diffeomorphisms of physical space: A particle starting at a point x at time 0 reaches a point y(x,t) at time t.  For each positive t, the function y(x) is one-to-one, onto, and differentiable. For incompressible fluids, the Lagrangian coordinate change yields a geodesic flow in a space of diffeomorphisms with an appropriate metric.

This talk reports on joint work with John Holmes and Feride Tiglay that carries out such a coordinate change for scalar conservation laws, such as Burgers equation, and systems of two conservation laws in one space dimension. We abstract the notion of a particle path and obtain a correspondence that is valid for weak solutions. We investigate the question of whether there can be an underlying geodesic flow for some metric.

Bio: Barbara Lee Keyfitz is Professor of Mathematics at the Ohio State University, which she joined in January 2009, after 21 years at the University of Houston and four and a half years as Director of the Fields Institute in Toronto, Canada.  She received her undergraduate education at the University of Toronto and her M.S. and Ph.D. from the Courant Institute, New York University.  Her research area is Nonlinear Partial Differential Equations.

Keyfitz is a SIAM Fellow, a Fellow of the American Association for the Advancement of Science, a Fellow of the American Mathematical Society, a Fellow of the Fields Institute, and the recipient of the 2012 SIAM Award for Distinguished Service to the Profession. She has served on the editorial boards of the AMS Proceedings, the AMS Transactions, JMAA, SIAM Journal of Applied Mathematics, Mathematical Methods in the Applied Sciences, Fields Institute Monographs and Communications, Chinese Journal of Engineering Mathematics, and as a member of the Mathematical Reviews Editorial Committee.  In 2012 she was the Noether Lecturer at the Joint Mathematics Meetings, and the Kovalevsky Lecturer at the SIAM Annual Meeting. She received the 2005 Krieger-Nelson Prize of the Canadian Mathematical Society, and an Honorary Doctor of Mathematics degree (2010) from the University of Waterloo.

Keyfitz was President of the Association for Women in Mathematics in 2005-2006, and was a Vice-President of SIAM, 1998-2003, and of the American Mathematical Society from 2011-2014. From 2011-2015 she served as President of the International Council on Industrial and Applied Mathematics.

Host: Tony Kearsley

Note: This talk will be recorded to provide access to NIST staff and associates who could not be present to the time of the seminar. The recording will be made available in the Math channel on NISTube, which is accessible only on the NIST internal network. This recording could be released to the public through a Freedom of Information Act (FOIA) request. Do not discuss or visually present any sensitive (CUI/PII/BII) material. Ensure that no inappropriate material or any minors are contained within the background of any recording. (To facilitate this, we request that cameras of attendees are muted except when asking questions.)

Created April 19, 2022, Updated May 27, 2022