Drs. Stefano Bianchini and Alberto Bressan will be honored for their breakthrough work in the analysis of partial differential equations, supplying new and powerful analytic ideas and establishing fundamental properties of the solutions.
Their paper, “Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems,” (Annals of Mathematics, 161 (2005), 223-342) has been selected for the SIAM Activity Group on Analysis of Partial Differential Equations Prize. The award will be presented on December 11 at the biennial SIAM Conference on Analysis of Partial Differential Equations, held this year in Mesa, Arizona. Bianchini will accept the prize and present their paper at a plenary lecture entitled “Singular Approximations to Hyperbolic Systems of Conservation Laws.”
Bianchini, of Italy’s International School of Advanced Studies, and Bressan, of Pennsylvania State University, derived a solution to a problem that has remained unsolved for 50 years: the question of identifying hyperbolic systems as idealized limits of systems with low viscosity, or resistance to flow.
“They solved the outstanding problem in the field, that is, to find the limit of the solutions as the viscosity tends to zero,” says Walter Strauss, Professor of Mathematics, Brown University. “They proved that the limit exists, is unique, and has finite total variation, at least if the initial variation is small. Their work has revolutionized this field of research.”
Nonlinear hyperbolic systems of partial differential equations are associated with a wide range of physical phenomena, for example, the dynamics of unsteady fluid motions, liquid-vapor flows, the evolution of astrophysical bodies, motion of gravitational waves, and, in materials science, the dynamics of interfaces between solids.
It has long been believed – without proof until now – that viscosity affects the system only in the microstructure of the discontinuities and boundary layers, while the gross properties of the flow – far from shocks and boundaries – are unchanged by the details of how viscosity is modeled.
“Bianchini and Bressan have found an existence theorem that demonstrates how solutions to hyperbolic conservation laws can be realized as limits of viscous systems,” says Barbara Lee Keyfitz, Professor of Mathematics at the University of Houston, and researcher in the field of nonlinear partial differential equations. “More important for the experts, they have been able to give a detailed analysis of how the wave structure actually evolves in the presence of viscosity. They are able to break a typical disturbance into its component parts, moving at different speeds, and to show that this evolution is independent of viscosity. It was the correct resolution of a wave into families—an idea that had eluded all researchers until now—that enabled this elegant result.”
The SIAM Activity Group on Analysis of Partial Differential Equations (SIAG/APDE) Prize, established in 2005, is awarded to the author(s) of the most outstanding paper published in the preceding four years, as determined by the prize committee. Members of the selection committee for the 2007 award were: Mary Pugh, Chair, University of Toronto, Canada; Yann Brenier, University of Nice, France; Alice Chang, Princeton University; Bjorn Engquist, University of Texas at Austin; Robert Pego, Carnegie Mellon University.
The Society for Industrial and Applied Mathematics (SIAM) is an international community of over 11,000 individual members, including applied and computational mathematicians, computer scientists, and other scientists and engineers. The Society advances the fields of applied mathematics and computational science by publishing a series of premier journals and a variety of books, and producing a wide selection of conferences. More information about SIAM is available at www.siam.org.
The SIAM Activity Group on Analysis of Partial Differential Equations fosters activity in the analysis of partial differential equations (PDE) and enhances communication between analysts, computational scientists and the broad PDE community. Its goals are to provide a forum where theoretical and applied researchers in the area can meet, to be an intellectual home for researchers in the analysis of PDE, to increase conference activity in PDE, and to enhance connections between SIAM and the mathematics community.