With an AWM-SIAM Sonia Kovalevsky Lecture titled “The Role of Characteristics in Conservation Laws,” Barbara Keyfitz emphasized her direct connection to the work of Sonia Kovalevsky (1850–1891), whose Cauchy–Kovalevsky theorem makes clear the importance of characteristics in PDEs. (At the Joint Math Meetings early this year, by contrast, Keyfitz titled her Noether Lecture “Conservation Laws—Not Exactly à la Noether,” pointing to the more distant relation between current research in conservation laws and the celebrated theorem of that name by Emmy Noether (1882–1935). In fact, Keyfitz says, it is well known that weak solutions to conservation laws do not preserve the symmetries described by Noether’s theorem; Noether herself moved away from her early work on invariant theory and went on to a career as a founder of modern algebra.)
The contributions in hyperbolic conservation laws for which the Kovalevsky committee cited Keyfitz include work with Herbert Kranzer on the “novel and important notion of singular (also called delta) shocks” and their properties. She and her collaborators also “spearheaded the revival of the rigourous treatment of transonic gas flow.” Her work has applications in aerodynamics and in models of multiphase flow in porous media.
In the lecture, Keyfitz followed an overview of progress in the field with a gratifying recent update: After “languishing for 25 years,” she said, one of her and Kranzer’s results was resuscitated recently by Marco Mazzotti of ETH Zurich. Mazzotti explored a variation on the familiar equations of two-component chromatography, which he terms “Langmuir/anti-Langmuir adsorption isotherm,” and discovered solutions that appeared in numerical simulations to exhibit singular shocks. In a real tour de force, Keyfitz said, Mazzotti and co-workers were able to design an experiment that demonstrated an example of these elusive objects.
“When I discovered numerically the singular solution in nonlinear binary chromatography,” Mazzotti wrote to SIAM News, “I had never heard of the existence of that type of solution to hyperbolic equations. Keyfitz’s papers allowed me to put my results into context. Particularly her 1995 paper with Kranzer (Journal of Differential Equations) was decisive in providing me with the conceptual and mathematical tools needed to find an explicit expression for the rate of propagation of the singular solution that matches nicely the experimental measurement. After learning of my first results and efforts, and despite my being a clear outsider, Barbara Keyfitz took the time to look into what I was doing and then encouraged me enthusiastically, thus providing very valuable support and motivation.”
One of Keyfitz’s points in the lecture was that characteristic surfaces give information on analytical as well as geometric properties of solutions of hyperbolic equations, via, for example, the symbol of the operator and the mechanism of Fourier transformation. This analysis is the basis of Rauch’s proof, following work of Brenner, that multidimensional conservation laws are unlikely to be well-posed in \(L_p\) unless \(p = 2\). The power of analysis of this type, she suggested, has not yet been fully exploited.
Partly as an illustration of related open questions, Keyfitz concluded with a computational result of which she was clearly very proud: an extension to the full Euler equations of ideal gas dynamics that exhibits (again numerically) a cascade of rarefaction waves, called “Guderley Mach reflection” by Keyfitz and collaborators Allen Tesdall and Richard Sanders (who built on pioneering work by Tesdall and John Hunter). This phenomenon has also been verified experimentally, she says, by Beric Skews and colleagues.
Peruse the gallery below for more highlights of the Annual Meeting.