Multidimensional Conservation LawsJune 12, 2006
The long, illustrious history of multidimensional conservation laws goes back to the founders of continuum mechanics, with Euler deserving the most recognition. Conservation laws are the familiar balance equations of mass, momentum, and energy, supplemented with an entropy inequality representing the second law of thermo-dynamics. Elasticity and gas dynamics provide the two most notable examples.
The study of these systems of equations as initial value problems in one space dimension became popular in the 1960s, propelled by the pioneering work of P. Lax, J. Glimm, and O. Oleinik, among others. A quick glance at the post-World War II literature, however, shows that the initial value problem was not always the dominant theme: The classic monographs of R. Courant and K.O. Friedrichs (Supersonic Flow and Shock Waves, 1948) and L. Bers (Mathematical Aspects of Subsonic and Transonic Flow, 1958), for example, pay scant (in fact no) attention to initial value problems. These books reflect the emphasis of the late 1940s and 1950s on boundary value problems for steady (or, in the case of self-similar solutions, pseudosteady) problems in gas dynamics. Here, of course, multidimensionality in space is of the essence, as in the flow of a compressible gas (air) around an airplane, or in the reflection of a shock wave from an obstacle.
While mathematical analysis of these steady multidimensional systems did not come to a complete halt after the 1950s (the work of C. Morawetz, B. Keyfitz, and S.-X. Chen and their collaborators being counterexamples), the area certainly saw nowhere near the advances made in the initial value problem in one space dimension. The difficulty, as is well known, is the change that can occur in the steady equations---from nice, well-understood elliptic equations to nasty, quasilinear hyperbolic systems across unknown free boundaries.
In an attempt to rectify the mathematical gap in the theory, a group of nine researchers at eight U.S. universities formed a National Science Foundation Focused Research Group (FRG), "Multidimensional Problems for the Euler Equations of Compressible Fluid Flow and Related Problems in Hyperbolic Conservation Laws." The members of the FRG are Suncica Canic, University of Houston; Gui-Qiang Chen, Northwestern University; Constantine Dafermos, Brown University; John Hunter, University of California, Davis; Tai-Ping Liu, Stanford University; Chi-Wang Shu, Brown University; Marshall Slemrod, University of Wisconsin, Madison; Dehua Wang, University of Pittsburgh; and Yuxi Zheng, Pennsylvania State University.
To date, the group has held four workshops (Pittsburgh, Pennsylvania, November 2003; Stanford University, July 2004; Madison, Wisconsin, June 2005; and Houston, Texas, March 2006); a fifth, organized by T.-P. Liu, is planned for June 2007, again at Stanford.
Statements of outstanding problems in both multidimensional initial and boundary value problems have been an important theme at all the workshops. There are, for example, absolutely no results on global time solutions to the initial value problem for the systems of gas dynamics in two and three space dimensions. Also lacking are results on existence of reflected shock waves for the self-similar Euler equations in two dimensions (Mach reflection). Each is a classic problem of striking physical importance for which mathematical undertanding is still an open issue.
Research results obtained by members of the group and others have been discussed at the workshops, as has the education of younger researchers about problems and techniques for approaching them. In just the past two or three years, impressive progress has been made on such problems as shock reflection, nozzle flow, flow through ducts, and flow over airfoils, and workshop participants have considered the importance of these advances in pointing the way to further results.
Interested readers can find a collection of papers by members of the FRG at http://www.math.pitt.edu/~dwang/FRG.html.
Marshall Slemrod (email@example.com) is a professor of mathematics at the University of Wisconsin, Madison.
NSF Focused Research Groups. The nine researchers who met periodically over the past three years to consider multidimensional conservation laws were supported as a National Science Foundation Focused Research Group (FRG) in the Mathematical Sciences. The FRG program allows "groups of researchers to respond to recognized scientific needs of pressing importance, to take advantage of current scientific opportunities, or to prepare the ground for anticipated significant scientific developments in the mathematical sciences." The program supports scientifically focused, well-delineated projects for which the collective effort by a group of researchers is necessary to reach the scientific goals.